Theorem 2euswap 1488

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 1086	. . . 4 |- (E.xE.yph -> E.yE.xph)
2	1	a1i 8	. . 3 |- (A.xE*yph -> (E.xE.yph -> E.yE.xph))
3		2moswap 1487	. . 3 |- (A.xE*yph -> (E*xE.yph -> E*yE.xph))
4	2, 3	anim12d 569	. 2 |- (A.xE*yph -> ((E.xE.yph /\ E*xE.yph) -> (E.yE.xph /\ E*yE.xph)))
5		eu5 1451	. 2 |- (E!xE.yph <-> (E.xE.yph /\ E*xE.yph))
6		eu5 1451	. 2 |- (E!yE.xph <-> (E.yE.xph /\ E*yE.xph))
7	4, 5, 6	3imtr4g 564	1 |- (A.xE*yph -> (E!xE.yph -> E!yE.xph))

Syntax hints:   -> wi 3   /\ wa 230  A.wal 995  E.wex 1021  E!weu 1422  E*wmo 1423

This theorem is referenced by:  euxfr2 1973  2reuswap 1984

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1003  ax-gen 1004  ax-8 1005  ax-10 1007  ax-11 1008  ax-12 1009  ax-17 1012  ax-4 1014  ax-5o 1016  ax-6o 1019  ax-9o 1164  ax-10o 1182  ax-16 1252  ax-11o 1260

This theorem depends on definitions:  df-bi 154  df-or 231  df-an 232  df-ex 1022  df-sb 1214  df-eu 1424  df-mo 1425

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