Theorem 2moswap 1064

Description: A condition allowing swap of "at most one" and existential quantifiers.

Assertion

Ref	Expression
2moswap	|- (A.xE*yph -> (E*xE.yph -> E*yE.xph))

Proof of Theorem 2moswap

Step	Hyp	Ref	Expression
1		hbe1 709	. . . . 5 |- (E.yph -> A.yE.yph)
2	1	moexex 1058	. . . 4 |- ((E*xE.yph /\ A.xE*yph) -> E*yE.x(E.yph /\ ph))
3	2	exp 291	. . 3 |- (E*xE.yph -> (A.xE*yph -> E*yE.x(E.yph /\ ph)))
4	3	com12 13	. 2 |- (A.xE*yph -> (E*xE.yph -> E*yE.x(E.yph /\ ph)))
5		19.8a 712	. . . . 5 |- (ph -> E.yph)
6	5	pm4.71ri 484	. . . 4 |- (ph <-> (E.yph /\ ph))
7	6	biex 733	. . 3 |- (E.xph <-> E.x(E.yph /\ ph))
8	7	bimo 1031	. 2 |- (E*yE.xph <-> E*yE.x(E.yph /\ ph))
9	4, 8	syl6ibr 186	1 |- (A.xE*yph -> (E*xE.yph -> E*yE.xph))

Syntax hints:   -> wi 2   /\ wa 196  A.wal 672  E.wex 678  E*wmo 1008

This theorem is referenced by:  2euswap 1065  2eu1 1067

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-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010

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