Theorem 19.12vv 960

Description: Special case of 19.12 729 where its converse holds.

Assertion

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

Distinct variable group(s):   x,y   ps,x   ph,y

Proof of Theorem 19.12vv

Step	Hyp	Ref	Expression
1		19.21v 942	. . 3 |- (A.y(ph -> ps) <-> (ph -> A.yps))
2	1	biex 733	. 2 |- (E.xA.y(ph -> ps) <-> E.x(ph -> A.yps))
3		19.36v 958	. 2 |- (E.x(ph -> A.yps) <-> (A.xph -> A.yps))
4		19.36v 958	. . . 4 |- (E.x(ph -> ps) <-> (A.xph -> ps))
5	4	bial 695	. . 3 |- (A.yE.x(ph -> ps) <-> A.y(A.xph -> ps))
6		19.21v 942	. . 3 |- (A.y(A.xph -> ps) <-> (A.xph -> A.yps))
7	5, 6	bitr2 152	. 2 |- ((A.xph -> A.yps) <-> A.yE.x(ph -> ps))
8	2, 3, 7	3bitr 155	1 |- (E.xA.y(ph -> ps) <-> A.yE.x(ph -> ps))

Syntax hints:   -> wi 2   <-> wb 127  A.wal 672  E.wex 678

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-gen 677  ax-17 925

This theorem depends on definitions:  df-bi 128  df-an 198  df-ex 679

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