Theorem 19.12b 31707

Description: Version of 19.12vv 2180 with not-free hypotheses, instead of distinct variable conditions. (Contributed by Scott Fenton, 13-Dec-2010.) (Revised by Mario Carneiro, 11-Dec-2016.)

Hypotheses

Ref	Expression
19.12b.1	|- F/ y ph
19.12b.2	|- F/ x ps

Assertion

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

Distinct variable group:   x, y

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

Proof of Theorem 19.12b

Step	Hyp	Ref	Expression
1		19.12b.1	. . . 4 |- F/ y ph
2	1	19.21 2075	. . 3 |- ( A. y ( ph -> ps ) <-> ( ph -> A. y ps ) )
3	2	exbii 1774	. 2 |- ( E. x A. y ( ph -> ps ) <-> E. x ( ph -> A. y ps ) )
4		19.12b.2	. . . 4 |- F/ x ps
5	4	nfal 2153	. . 3 |- F/ x A. y ps
6	5	19.36 2098	. 2 |- ( E. x ( ph -> A. y ps ) <-> ( A. x ph -> A. y ps ) )
7	4	19.36 2098	. . . 4 |- ( E. x ( ph -> ps ) <-> ( A. x ph -> ps ) )
8	7	albii 1747	. . 3 |- ( A. y E. x ( ph -> ps ) <-> A. y ( A. x ph -> ps ) )
9	1	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	8, 10	bitr2i 265	. 2 |- ( ( A. x ph -> A. y ps ) <-> A. y E. x ( ph -> ps ) )
12	3, 6, 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   F/wnf 1708

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)