Theorem 19.21 1960

Description: Theorem 19.21 of [Margaris] p. 90. The hypothesis can be thought of as " x is not free in ph." See 19.21v 1775 for a version requiring fewer axioms. See also 19.21h 1962. (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 1959	. 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 187   A.wal 1435   F/wnf 1663

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1839  ax-10 1887  ax-12 1905

This theorem depends on definitions:  df-bi 188  df-ex 1660  df-nf 1664

This theorem is referenced by:  19.21-2  1961  19.21h  1962  stdpc5  1963  nf3  2017  19.32  2022  19.21vOLD  2036  19.12vv  2041  cbv1  2071  axc14  2166  r2alf  2801  r2alfOLD  2802  19.12b  30443  bj-biexal2  31248  bj-bialal  31250  bj-cbv1v  31280  wl-dral1d  31778  mpt2bi123f  32320