Theorem 2euswap 1849

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 1392	. . . 4 |- (E.xE.yph -> E.yE.xph)
2	1	a1i 8	. . 3 |- (A.xE*yph -> (E.xE.yph -> E.yE.xph))
3		2moswap 1848	. . 3 |- (A.xE*yph -> (E*xE.yph -> E*yE.xph))
4	2, 3	anim12d 617	. 2 |- (A.xE*yph -> ((E.xE.yph /\ E*xE.yph) -> (E.yE.xph /\ E*yE.xph)))
5		eu5 1805	. 2 |- (E!xE.yph <-> (E.xE.yph /\ E*xE.yph))
6		eu5 1805	. 2 |- (E!yE.xph <-> (E.yE.xph /\ E*yE.xph))
7	4, 5, 6	3imtr4g 612	1 |- (A.xE*yph -> (E!xE.yph -> E!yE.xph))

Syntax hints:   -> wi 3   /\ wa 240  A.wal 1296  E.wex 1326  E!weu 1771  E*wmo 1772

This theorem is referenced by:  euxfr2 2437  2reuswap 2449

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776

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