Theorem 19.21 1844

Description: Theorem 19.21 of [Margaris] p. 90. The hypothesis can be thought of as " x is not free in ph." (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 1843	. 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 1368   F/wnf 1590

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-12 1794

This theorem depends on definitions:  df-bi 185  df-ex 1588  df-nf 1591

This theorem is referenced by:  19.21h  1845  stdpc5  1846  19.21-2  1898  nf3  1901  19.32  1907  19.21v  1921  19.12vv  1929  cbv1  1973  axc14  2073  eu2OLD  2316  moanimOLD  2342  r2alf  2875  19.12b  27760  wl-dral1d  28509  mpt2bi123f  29124  bj-biexal2  32529  bj-bialal  32531  bj-cbv1v  32561