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Theorem cla4gf 1851
Description: Rule of specialization with implicit substitution. Compare Theorem 7.3 of [Quine] p. 44.
Hypotheses
Ref Expression
cla4gf.1 |- (y e. A -> A.x y e. A)
cla4gf.2 |- (ps -> A.xps)
cla4gf.3 |- (x = A -> (ph <-> ps))
Assertion
Ref Expression
cla4gf |- (A e. B -> (A.xph -> ps))
Distinct variable groups:   x,y   y,A
Proof of Theorem cla4gf
StepHypRef Expression
1 elisset 1808 . 2 |- (A e. B -> A e. V)
2 isset 1805 . . . . 5 |- (A e. V <-> E.y y = A)
3 cla4gf.1 . . . . . . 7 |- (y e. A -> A.x y e. A)
43hbeleq 1559 . . . . . 6 |- (y = A -> A.x y = A)
5 ax-17 968 . . . . . 6 |- (x = A -> A.y x = A)
6 eqeq1 1473 . . . . . 6 |- (y = x -> (y = A <-> x = A))
74, 5, 6cbvex 1162 . . . . 5 |- (E.y y = A <-> E.x x = A)
82, 7bitr 173 . . . 4 |- (A e. V <-> E.x x = A)
9 cla4gf.3 . . . . . 6 |- (x = A -> (ph <-> ps))
109biimpd 153 . . . . 5 |- (x = A -> (ph -> ps))
111019.22i 1036 . . . 4 |- (E.x x = A -> E.x(ph -> ps))
128, 11sylbi 199 . . 3 |- (A e. V -> E.x(ph -> ps))
13 cla4gf.2 . . . 4 |- (ps -> A.xps)
141319.36 1074 . . 3 |- (E.x(ph -> ps) <-> (A.xph -> ps))
1512, 14sylib 198 . 2 |- (A e. V -> (A.xph -> ps))
161, 15syl 10 1 |- (A e. B -> (A.xph -> ps))
Colors of variables: wff set class
Syntax hints:   -> wi 3   <-> wb 146  A.wal 951   = wceq 953   e. wcel 955  E.wex 977  Vcvv 1802
This theorem is referenced by:  cla4egf 1852  cla4gv 1853  rcla4 1862  moi 1915
This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-12 965  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-ext 1452
This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 978  df-sb 1168  df-clab 1457  df-cleq 1462  df-clel 1465  df-v 1803
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