Theorem 19.21-2 1660

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 1576	. . 3 |- ( A. y ( ph -> ps ) <-> ( ph -> A. y ps ) )
3	2	albii 1463	. 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 1576	. 2 |- ( A. x ( ph -> A. y ps ) <-> ( ph -> A. x A. y ps ) )
6	3, 5	bitri 183	1 |- ( A. x A. y ( ph -> ps ) <-> ( ph -> A. x A. y ps ) )

Syntax hints:   -> wi 4   <-> wb 104   A.wal 1346   F/wnf 1453

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1440  ax-gen 1442  ax-4 1503  ax-ial 1527  ax-i5r 1528

This theorem depends on definitions:  df-bi 116  df-nf 1454

This theorem is referenced by: (None)