Theorem rmo2i 3041

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 2512	. 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 3040	. 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 1343   F/wnf 1448   E.wex 1480   A.wral 2444   E.wrex 2445   E*wrmo 2447

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523

This theorem depends on definitions:  df-bi 116  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-ral 2449  df-rex 2450  df-rmo 2452

This theorem is referenced by: (None)