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

Step	Hyp	Ref	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 ) ) )
3	1, 2	ax-mp 5	1 |- ( A. x ( ph -> ps ) <-> ( ph -> A. x ps ) )

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