Theorem 2moswap 1437

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 1012	. . . 4 |- (E.yph -> A.yE.yph)
2	1	moexex 1431	. . 3 |- ((E*xE.yph /\ A.xE*yph) -> E*yE.x(E.yph /\ ph))
3	2	expcom 374	. 2 |- (A.xE*yph -> (E*xE.yph -> E*yE.x(E.yph /\ ph)))
4		19.8a 1025	. . . . 5 |- (ph -> E.yph)
5	4	pm4.71ri 636	. . . 4 |- (ph <-> (E.yph /\ ph))
6	5	exbii 1047	. . 3 |- (E.xph <-> E.x(E.yph /\ ph))
7	6	mobii 1398	. 2 |- (E*yE.xph <-> E*yE.x(E.yph /\ ph))
8	3, 7	syl6ibr 213	1 |- (A.xE*yph -> (E*xE.yph -> E*yE.xph))

Syntax hints:   -> wi 3   /\ wa 223  A.wal 951  E.wex 977  E*wmo 1374

This theorem is referenced by:  2euswap 1438  2eu1 1442

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-10 963  ax-11 964  ax-12 965  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 978  df-sb 1168  df-eu 1375  df-mo 1376

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