Theorem 19.21-2 1557

Description: Theorem 19.21 of [Margaris] p. 90 but with 2 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 1475	. . 3 |- ( A. y ( ph -> ps ) <-> ( ph -> A. y ps ) )
3	2	albii 1359	. 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 1475	. 2 |- ( A. x ( ph -> A. y ps ) <-> ( ph -> A. x A. y ps ) )
6	3, 5	bitri 173	1 |- ( A. x A. y ( ph -> ps ) <-> ( ph -> A. x A. y ps ) )

Syntax hints:   -> wi 4   <-> wb 98   A.wal 1241   F/wnf 1349

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-gen 1338  ax-4 1400  ax-ial 1427  ax-i5r 1428

This theorem depends on definitions:  df-bi 110  df-nf 1350

This theorem is referenced by: (None)