Theorem 2euswap 1485

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 1081	. . . 4 |- (E.xE.yph -> E.yE.xph)
2	1	a1i 8	. . 3 |- (A.xE*yph -> (E.xE.yph -> E.yE.xph))
3		2moswap 1484	. . 3 |- (A.xE*yph -> (E*xE.yph -> E*yE.xph))
4	2, 3	anim12d 561	. 2 |- (A.xE*yph -> ((E.xE.yph /\ E*xE.yph) -> (E.yE.xph /\ E*yE.xph)))
5		eu5 1448	. 2 |- (E!xE.yph <-> (E.xE.yph /\ E*xE.yph))
6		eu5 1448	. 2 |- (E!yE.xph <-> (E.yE.xph /\ E*yE.xph))
7	4, 5, 6	3imtr4g 556	1 |- (A.xE*yph -> (E!xE.yph -> E!yE.xph))

Syntax hints:   -> wi 3   /\ wa 221  A.wal 990  E.wex 1016  E!weu 1419  E*wmo 1420

This theorem is referenced by:  euxfr2 1972  2reuswap 1983

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 998  ax-gen 999  ax-8 1000  ax-10 1002  ax-11 1003  ax-12 1004  ax-17 1007  ax-4 1009  ax-5o 1011  ax-6o 1014  ax-9o 1159  ax-10o 1177  ax-16 1247  ax-11o 1255

This theorem depends on definitions:  df-bi 145  df-or 222  df-an 223  df-ex 1017  df-sb 1209  df-eu 1421  df-mo 1422

Copyright terms: Public domain