Theorem 19.21-2 2078

Description: Version of 19.21 2075 with two quantifiers. (Contributed by NM, 4-Feb-2005.)

Hypotheses

Ref	Expression
19.21-2.1	|- F/ x ph
19.21-2.2	|- F/ y ph

Assertion

Ref	Expression
19.21-2	|- ( A. x A. y ( ph -> ps ) <-> ( ph -> A. x A. y ps ) )

Proof of Theorem 19.21-2

Step	Hyp	Ref	Expression
1		19.21-2.2	. . . 4 |- F/ y ph
2	1	19.21 2075	. . 3 |- ( A. y ( ph -> ps ) <-> ( ph -> A. y ps ) )
3	2	albii 1747	. 2 |- ( A. x A. y ( ph -> ps ) <-> A. x ( ph -> A. y ps ) )
4		19.21-2.1	. . 3 |- F/ x ph
5	4	19.21 2075	. 2 |- ( A. x ( ph -> A. y ps ) <-> ( ph -> A. x A. y ps ) )
6	3, 5	bitri 264	1 |- ( A. x A. y ( ph -> ps ) <-> ( ph -> A. x A. y 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:  cotr2g  13715  dford4  37596