Theorem 19.12vv 1297

Description: Special case of 19.12 1043 where its converse holds.

Assertion

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

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

Proof of Theorem 19.12vv

Step	Hyp	Ref	Expression
1		19.21v 1280	. . 3 |- (A.y(ph -> ps) <-> (ph -> A.yps))
2	1	exbii 1047	. 2 |- (E.xA.y(ph -> ps) <-> E.x(ph -> A.yps))
3		19.36v 1295	. 2 |- (E.x(ph -> A.yps) <-> (A.xph -> A.yps))
4		19.36v 1295	. . . 4 |- (E.x(ph -> ps) <-> (A.xph -> ps))
5	4	albii 996	. . 3 |- (A.yE.x(ph -> ps) <-> A.y(A.xph -> ps))
6		19.21v 1280	. . 3 |- (A.y(A.xph -> ps) <-> (A.xph -> A.yps))
7	5, 6	bitr2 174	. 2 |- ((A.xph -> A.yps) <-> A.yE.x(ph -> ps))
8	2, 3, 7	3bitr 177	1 |- (E.xA.y(ph -> ps) <-> A.yE.x(ph -> ps))

Syntax hints:   -> wi 3   <-> wb 146  A.wal 951  E.wex 977

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-gen 960  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 978

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