Theorem 2moswap 2279

Description: A condition allowing swap of "at most one" and existential quantifiers. (Contributed by NM, 10-Apr-2004.)

Assertion

Ref	Expression
2moswap	|- (A.xE*yph -> (E*xE.yph -> E*yE.xph))

Proof of Theorem 2moswap

Step	Hyp	Ref	Expression
1		nfe1 1732	. . . 4 |- F/yE.yph
2	1	moexex 2273	. . 3 |- ((E*xE.yph /\ A.xE*yph) -> E*yE.x(E.yph /\ ph))
3	2	expcom 424	. 2 |- (A.xE*yph -> (E*xE.yph -> E*yE.x(E.yph /\ ph)))
4		19.8a 1756	. . . . 5 |- (ph -> E.yph)
5	4	pm4.71ri 614	. . . 4 |- (ph <-> (E.yph /\ ph))
6	5	exbii 1582	. . 3 |- (E.xph <-> E.x(E.yph /\ ph))
7	6	mobii 2240	. 2 |- (E*yE.xph <-> E*yE.x(E.yph /\ ph))
8	3, 7	syl6ibr 218	1 |- (A.xE*yph -> (E*xE.yph -> E*yE.xph))

Syntax hints:   -> wi 4   /\ wa 358  A.wal 1540  E.wex 1541  E*wmo 2205

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209

This theorem is referenced by:  2euswap  2280  2eu1  2284