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Theorem 19.21h 1494
Description: Theorem 19.21 of [Margaris] p. 90. The hypothesis can be thought of as " x is not free in  ph." New proofs should use 19.21 1520 instead. (Contributed by NM, 5-Aug-1993.) (New usage is discouraged.)
Hypothesis
Ref Expression
19.21h.1  |-  ( ph  ->  A. x ph )
Assertion
Ref Expression
19.21h  |-  ( A. x ( ph  ->  ps )  <->  ( ph  ->  A. x ps ) )

Proof of Theorem 19.21h
StepHypRef Expression
1 19.21h.1 . . 3  |-  ( ph  ->  A. x ph )
2 alim 1391 . . 3  |-  ( A. x ( ph  ->  ps )  ->  ( A. x ph  ->  A. x ps ) )
31, 2syl5 32 . 2  |-  ( A. x ( ph  ->  ps )  ->  ( ph  ->  A. x ps )
)
4 hba1 1478 . . . 4  |-  ( A. x ps  ->  A. x A. x ps )
51, 4hbim 1482 . . 3  |-  ( (
ph  ->  A. x ps )  ->  A. x ( ph  ->  A. x ps )
)
6 ax-4 1445 . . . 4  |-  ( A. x ps  ->  ps )
76imim2i 12 . . 3  |-  ( (
ph  ->  A. x ps )  ->  ( ph  ->  ps ) )
85, 7alrimih 1403 . 2  |-  ( (
ph  ->  A. x ps )  ->  A. x ( ph  ->  ps ) )
93, 8impbii 124 1  |-  ( A. x ( ph  ->  ps )  <->  ( ph  ->  A. x ps ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 103   A.wal 1287
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia2 105  ax-ia3 106  ax-5 1381  ax-gen 1383  ax-4 1445  ax-ial 1472  ax-i5r 1473
This theorem depends on definitions:  df-bi 115
This theorem is referenced by:  hbim1  1507  nf3  1604  19.21v  1801
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