Theorem pm11.53v 1906

Description: Version of pm11.53 2179 with a dv condition, requiring fewer axioms. (Contributed by BJ, 7-Mar-2020.)

Assertion

Ref	Expression
pm11.53v	|- ( A. x A. y ( ph -> ps ) <-> ( E. x ph -> A. y ps ) )

Distinct variable groups:   ph, y   ps, x   x, y

Allowed substitution hints:   ph( x)   ps( y)

Proof of Theorem pm11.53v

Step	Hyp	Ref	Expression
1		19.21v 1868	. . 3 |- ( A. y ( ph -> ps ) <-> ( ph -> A. y ps ) )
2	1	albii 1747	. 2 |- ( A. x A. y ( ph -> ps ) <-> A. x ( ph -> A. y ps ) )
3		19.23v 1902	. 2 |- ( A. x ( ph -> A. y ps ) <-> ( E. x ph -> A. y ps ) )
4	2, 3	bitri 264	1 |- ( A. x A. y ( ph -> ps ) <-> ( E. x ph -> A. y ps ) )

Syntax hints:   -> wi 4   <-> wb 196   A.wal 1481   E.wex 1704

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

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

This theorem is referenced by:  sbnf2  2439