Theorem rmo2i 3097

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 2554	. 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 3096	. 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 1373   F/wnf 1484   E.wex 1516   A.wral 2486   E.wrex 2487   E*wrmo 2489

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559

This theorem depends on definitions:  df-bi 117  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-ral 2491  df-rex 2492  df-rmo 2494

This theorem is referenced by: (None)