Theorem 19.12vvv 1907

Description: Version of 19.12vv 2180 with a dv condition, requiring fewer axioms. See also 19.12 2164. (Contributed by BJ, 18-Mar-2020.)

Assertion

Ref	Expression
19.12vvv	|- ( E. x A. y ( ph -> ps ) <-> A. y E. x ( ph -> ps ) )

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

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

Proof of Theorem 19.12vvv

Step	Hyp	Ref	Expression
1		19.21v 1868	. . 3 |- ( A. y ( ph -> ps ) <-> ( ph -> A. y ps ) )
2	1	exbii 1774	. 2 |- ( E. x A. y ( ph -> ps ) <-> E. x ( ph -> A. y ps ) )
3		19.36v 1904	. 2 |- ( E. x ( ph -> A. y ps ) <-> ( A. x ph -> A. y ps ) )
4		19.36v 1904	. . . 4 |- ( E. x ( ph -> ps ) <-> ( A. x ph -> ps ) )
5	4	albii 1747	. . 3 |- ( A. y E. x ( ph -> ps ) <-> A. y ( A. x ph -> ps ) )
6		19.21v 1868	. . 3 |- ( A. y ( A. x ph -> ps ) <-> ( A. x ph -> A. y ps ) )
7	5, 6	bitr2i 265	. 2 |- ( ( A. x ph -> A. y ps ) <-> A. y E. x ( ph -> ps ) )
8	2, 3, 7	3bitri 286	1 |- ( E. x A. y ( ph -> ps ) <-> A. y E. x ( ph -> 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: (None)