Theorem rmo2i 3134

Description: Condition implying restricted at-most-one quantifier. (Contributed by NM, 17-Jun-2017.)

Hypothesis

Ref	Expression
rmo2.1	|- F/ y ph

Assertion

Ref	Expression
rmo2i	|- ( E. y e. A A. x e. A ( ph -> x = y ) -> E* x e. A ph )

Distinct variable group:   x, y, A

Allowed substitution hints:   ph( x, y)

Proof of Theorem rmo2i

Step	Hyp	Ref	Expression
1		rexex 2588	. 2 |- ( E. y e. A A. x e. A ( ph -> x = y ) -> E. y A. x e. A ( ph -> x = y ) )
2		rmo2.1	. . 3 |- F/ y ph
3	2	rmo2ilem 3133	. 2 |- ( E. y A. x e. A ( ph -> x = y ) -> E* x e. A ph )
4	1, 3	syl 14	1 |- ( E. y e. A A. x e. A ( ph -> x = y ) -> E* x e. A ph )

Syntax hints:   -> wi 4   = wceq 1398   F/wnf 1509   E.wex 1541   A.wral 2520   E.wrex 2521   E*wrmo 2523

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584

This theorem depends on definitions:  df-bi 117  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-ral 2525  df-rex 2526  df-rmo 2528

This theorem is referenced by: (None)