Theorem 2moswap 1848

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 1363	. . . 4 |- (E.yph -> A.yE.yph)
2	1	moexex 1841	. . 3 |- ((E*xE.yph /\ A.xE*yph) -> E*yE.x(E.yph /\ ph))
3	2	expcom 403	. 2 |- (A.xE*yph -> (E*xE.yph -> E*yE.x(E.yph /\ ph)))
4		19.8a 1376	. . . . 5 |- (ph -> E.yph)
5	4	pm4.71ri 700	. . . 4 |- (ph <-> (E.yph /\ ph))
6	5	exbii 1398	. . 3 |- (E.xph <-> E.x(E.yph /\ ph))
7	6	mobii 1801	. 2 |- (E*yE.xph <-> E*yE.x(E.yph /\ ph))
8	3, 7	syl6ibr 230	1 |- (A.xE*yph -> (E*xE.yph -> E*yE.xph))

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

This theorem is referenced by:  2euswap 1849  2eu1 1853

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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