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Theorem 19.23t 1114
Description: Closed form of Theorem 19.23 of [Margaris] p. 90.
Assertion
Ref Expression
19.23t |- (A.x(ps -> A.xps) -> (A.x(ph -> ps) <-> (E.xph -> ps)))
Proof of Theorem 19.23t
StepHypRef Expression
1 hba1 1001 . . 3 |- (A.x(ps -> A.xps) -> A.xA.x(ps -> A.xps))
2 ax-4 971 . . . . 5 |- (A.xps -> ps)
3 ax-4 971 . . . . 5 |- (A.x(ps -> A.xps) -> (ps -> A.xps))
42, 3impbid2 517 . . . 4 |- (A.x(ps -> A.xps) -> (A.xps <-> ps))
54imbi2d 611 . . 3 |- (A.x(ps -> A.xps) -> ((ph -> A.xps) <-> (ph -> ps)))
61, 5albid 1102 . 2 |- (A.x(ps -> A.xps) -> (A.x(ph -> A.xps) <-> A.x(ph -> ps)))
74imbi2d 611 . . 3 |- (A.x(ps -> A.xps) -> ((E.xph -> A.xps) <-> (E.xph -> ps)))
8 hba1 1001 . . . 4 |- (A.xps -> A.xA.xps)
9819.23 1061 . . 3 |- (A.x(ph -> A.xps) <-> (E.xph -> A.xps))
107, 9syl5bb 531 . 2 |- (A.x(ps -> A.xps) -> (A.x(ph -> A.xps) <-> (E.xph -> ps)))
116, 10bitr3d 529 1 |- (A.x(ps -> A.xps) -> (A.x(ph -> ps) <-> (E.xph -> ps)))
Colors of variables: wff set class
Syntax hints:   -> wi 3   <-> wb 146  A.wal 952  E.wex 978
This theorem is referenced by:  vtoclegft 1852  sbciegft 1955
This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-gen 961  ax-4 971  ax-5o 973  ax-6o 976
This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 979
Copyright terms: Public domain