Theorem 19.21h 2121

Description: Theorem 19.21 of [Margaris] p. 90. The hypothesis can be thought of as " x is not free in ph." See also 19.21 2075 and 19.21v 1868. (Contributed by NM, 1-Aug-2017.) (Proof shortened by Wolf Lammen, 1-Jan-2018.)

Hypothesis

Ref	Expression
19.21h.1	|- ( ph -> A. x ph )

Assertion

Ref	Expression
19.21h	|- ( A. x ( ph -> ps ) <-> ( ph -> A. x ps ) )

Proof of Theorem 19.21h

Step	Hyp	Ref	Expression
1		19.21h.1	. . 3 |- ( ph -> A. x ph )
2	1	nf5i 2024	. 2 |- F/ x ph
3	2	19.21 2075	1 |- ( A. x ( ph -> ps ) <-> ( ph -> A. x ps ) )

Syntax hints:   -> wi 4   <-> wb 196   A.wal 1481

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-10 2019  ax-12 2047

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

This theorem is referenced by:  hbim1  2125