Theorem 2moswap 1487

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 1057	. . . 4 |- (E.yph -> A.yE.yph)
2	1	moexex 1481	. . 3 |- ((E*xE.yph /\ A.xE*yph) -> E*yE.x(E.yph /\ ph))
3	2	expcom 381	. 2 |- (A.xE*yph -> (E*xE.yph -> E*yE.x(E.yph /\ ph)))
4		19.8a 1070	. . . . 5 |- (ph -> E.yph)
5	4	pm4.71ri 649	. . . 4 |- (ph <-> (E.yph /\ ph))
6	5	exbii 1092	. . 3 |- (E.xph <-> E.x(E.yph /\ ph))
7	6	mobii 1447	. 2 |- (E*yE.xph <-> E*yE.x(E.yph /\ ph))
8	3, 7	syl6ibr 220	1 |- (A.xE*yph -> (E*xE.yph -> E*yE.xph))

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

This theorem is referenced by:  2euswap 1488  2eu1 1492

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

Copyright terms: Public domain