Theorem 19.21 2075

Description: Theorem 19.21 of [Margaris] p. 90. The hypothesis can be thought of as " x is not free in ph." See 19.21v 1868 for a version requiring fewer axioms. See also 19.21h 2121. (Contributed by NM, 14-May-1993.) (Revised by Mario Carneiro, 24-Sep-2016.) df-nf 1710 changed. (Revised by Wolf Lammen, 18-Sep-2021.)

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 2073	. 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 196   A.wal 1481   F/wnf 1708

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-12 2047

This theorem depends on definitions:  df-bi 197  df-ex 1705  df-nf 1710

This theorem is referenced by:  stdpc5  2076  19.21-2  2078  19.32  2101  nf6  2117  19.21h  2121  19.12vv  2180  cbv1  2267  axc14  2372  r2alf  2938  19.12b  31707  bj-biexal2  32697  bj-bialal  32699  bj-cbv1v  32729  wl-dral1d  33318  mpt2bi123f  33971