Theorem eeor 1120

Description: Rearrange existential quantifiers.

Hypotheses

Ref	Expression
eeor.1	|- (ph -> A.yph)
eeor.2	|- (ps -> A.xps)

Assertion

Ref	Expression
eeor	|- (E.xE.y(ph \/ ps) <-> (E.xph \/ E.yps))

Proof of Theorem eeor

Step	Hyp	Ref	Expression
1		eeor.1	. . . 4 |- (ph -> A.yph)
2	1	19.45 1090	. . 3 |- (E.y(ph \/ ps) <-> (ph \/ E.yps))
3	2	exbii 1051	. 2 |- (E.xE.y(ph \/ ps) <-> E.x(ph \/ E.yps))
4		eeor.2	. . . 4 |- (ps -> A.xps)
5	4	hbex 1006	. . 3 |- (E.yps -> A.xE.yps)
6	5	19.44 1089	. 2 |- (E.x(ph \/ E.yps) <-> (E.xph \/ E.yps))
7	3, 6	bitr 173	1 |- (E.xE.y(ph \/ ps) <-> (E.xph \/ E.yps))

Syntax hints:   -> wi 3   <-> wb 146   \/ wo 222  A.wal 954  E.wex 980

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-4 973  ax-5o 975  ax-6o 978

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 981

Copyright terms: Public domain