Theorem 19.12vv 2180

Description: Special case of 19.12 2164 where its converse holds. See 19.12vvv 1907 for a version with a dv condition requiring fewer axioms. (Contributed by NM, 18-Jul-2001.) (Revised by Andrew Salmon, 11-Jul-2011.)

Assertion

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

Distinct variable groups:   ps, x   ph, y

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

Proof of Theorem 19.12vv

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		nfv 1843	. . . 4 |- F/ x ps
4	3	nfal 2153	. . 3 |- F/ x A. y ps
5	4	19.36 2098	. 2 |- ( E. x ( ph -> A. y ps ) <-> ( A. x ph -> A. y ps ) )
6		19.36v 1904	. . . 4 |- ( E. x ( ph -> ps ) <-> ( A. x ph -> ps ) )
7	6	albii 1747	. . 3 |- ( A. y E. x ( ph -> ps ) <-> A. y ( A. x ph -> ps ) )
8		nfv 1843	. . . . 5 |- F/ y ph
9	8	nfal 2153	. . . 4 |- F/ y A. x ph
10	9	19.21 2075	. . 3 |- ( A. y ( A. x ph -> ps ) <-> ( A. x ph -> A. y ps ) )
11	7, 10	bitr2i 265	. 2 |- ( ( A. x ph -> A. y ps ) <-> A. y E. x ( ph -> ps ) )
12	2, 5, 11	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  ax-7 1935  ax-10 2019  ax-11 2034  ax-12 2047

This theorem depends on definitions:  df-bi 197  df-ex 1705  df-nf 1710

This theorem is referenced by: (None)