Theorem 2moswap 1484

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 1052	. . . 4 |- (E.yph -> A.yE.yph)
2	1	moexex 1478	. . 3 |- ((E*xE.yph /\ A.xE*yph) -> E*yE.x(E.yph /\ ph))
3	2	expcom 372	. 2 |- (A.xE*yph -> (E*xE.yph -> E*yE.x(E.yph /\ ph)))
4		19.8a 1065	. . . . 5 |- (ph -> E.yph)
5	4	pm4.71ri 641	. . . 4 |- (ph <-> (E.yph /\ ph))
6	5	exbii 1087	. . 3 |- (E.xph <-> E.x(E.yph /\ ph))
7	6	mobii 1444	. 2 |- (E*yE.xph <-> E*yE.x(E.yph /\ ph))
8	3, 7	syl6ibr 211	1 |- (A.xE*yph -> (E*xE.yph -> E*yE.xph))

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

This theorem is referenced by:  2euswap 1485  2eu1 1489

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