Theorem rmo2i 3054

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 2523	. 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 3053	. 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 1353   F/wnf 1460   E.wex 1492   A.wral 2455   E.wrex 2456   E*wrmo 2458

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535

This theorem depends on definitions:  df-bi 117  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-ral 2460  df-rex 2461  df-rmo 2463

This theorem is referenced by: (None)