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

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

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