Theorem 2euswap 1438

Description: A condition allowing swap of uniqueness and existential quantifiers.

Assertion

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

Proof of Theorem 2euswap

Step	Hyp	Ref	Expression
1		excomim 1041	. . . 4 |- (E.xE.yph -> E.yE.xph)
2	1	a1i 8	. . 3 |- (A.xE*yph -> (E.xE.yph -> E.yE.xph))
3		2moswap 1437	. . 3 |- (A.xE*yph -> (E*xE.yph -> E*yE.xph))
4	2, 3	anim12d 556	. 2 |- (A.xE*yph -> ((E.xE.yph /\ E*xE.yph) -> (E.yE.xph /\ E*yE.xph)))
5		eu5 1402	. 2 |- (E!xE.yph <-> (E.xE.yph /\ E*xE.yph))
6		eu5 1402	. 2 |- (E!yE.xph <-> (E.yE.xph /\ E*yE.xph))
7	4, 5, 6	3imtr4g 551	1 |- (A.xE*yph -> (E!xE.yph -> E!yE.xph))

Syntax hints:   -> wi 3   /\ wa 223  A.wal 951  E.wex 977  E!weu 1373  E*wmo 1374

This theorem is referenced by:  euxfr2 1916  2reuswap 1927

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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