Theorem 2euswap 2280

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

Assertion

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

Proof of Theorem 2euswap

Step	Hyp	Ref	Expression
1		excomim 1742	. . . 4 |- (E.xE.yph -> E.yE.xph)
2	1	a1i 10	. . 3 |- (A.xE*yph -> (E.xE.yph -> E.yE.xph))
3		2moswap 2279	. . 3 |- (A.xE*yph -> (E*xE.yph -> E*yE.xph))
4	2, 3	anim12d 546	. 2 |- (A.xE*yph -> ((E.xE.yph /\ E*xE.yph) -> (E.yE.xph /\ E*yE.xph)))
5		eu5 2242	. 2 |- (E!xE.yph <-> (E.xE.yph /\ E*xE.yph))
6		eu5 2242	. 2 |- (E!yE.xph <-> (E.yE.xph /\ E*yE.xph))
7	4, 5, 6	3imtr4g 261	1 |- (A.xE*yph -> (E!xE.yph -> E!yE.xph))

Syntax hints:   -> wi 4   /\ wa 358  A.wal 1540  E.wex 1541  E!weu 2204  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:  euxfr2  3022  2reuswap  3039