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Theorem 19.21 1910
Description: Theorem 19.21 of [Margaris] p. 90. The hypothesis can be thought of as " x is not free in  ph." See 19.21v 1734 for a version requiring fewer axioms. See also 19.21h 1912. (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 1909 . 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 184   A.wal 1396   F/wnf 1621
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-10 1842  ax-12 1859
This theorem depends on definitions:  df-bi 185  df-ex 1618  df-nf 1622
This theorem is referenced by:  19.21-2  1911  19.21h  1912  stdpc5  1913  nf3  1966  19.32  1972  19.21vOLD  1986  19.12vv  1991  cbv1  2022  axc14  2115  r2alf  2830  r2alfOLD  2831  19.12b  29474  wl-dral1d  30224  mpt2bi123f  30811  bj-biexal2  34661  bj-bialal  34663  bj-cbv1v  34693
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