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Theorem 19.21 1987
Description: Theorem 19.21 of [Margaris] p. 90. The hypothesis can be thought of as " x is not free in  ph." See 19.21v 1786 for a version requiring fewer axioms. See also 19.21h 1989. (Contributed by NM, 14-May-1993.) (Revised by Mario Carneiro, 24-Sep-2016.)
Hypothesis
Ref Expression
19.21.1  |-  F/ x ph
Assertion
Ref Expression
19.21  |-  ( A. x ( ph  ->  ps )  <->  ( ph  ->  A. x ps ) )

Proof of Theorem 19.21
StepHypRef Expression
1 19.21.1 . 2  |-  F/ x ph
2 19.21t 1986 . 2  |-  ( F/ x ph  ->  ( A. x ( ph  ->  ps )  <->  ( ph  ->  A. x ps ) ) )
31, 2ax-mp 5 1  |-  ( A. x ( ph  ->  ps )  <->  ( ph  ->  A. x ps ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 188   A.wal 1442   F/wnf 1667
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-10 1915  ax-12 1933
This theorem depends on definitions:  df-bi 189  df-an 373  df-ex 1664  df-nf 1668
This theorem is referenced by:  19.21-2  1988  19.21h  1989  stdpc5  1990  nf3  2042  19.32  2047  19.21vOLD  2071  19.12vv  2076  cbv1  2110  axc14  2201  r2alf  2764  r2alfOLD  2765  19.12b  30448  bj-biexal2  31300  bj-bialal  31302  bj-cbv1v  31330  wl-dral1d  31864  mpt2bi123f  32406
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