Theorem rmo2i 3068

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 2536	. 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 3067	. 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 1364   F/wnf 1471   E.wex 1503   A.wral 2468   E.wrex 2469   E*wrmo 2471

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546

This theorem depends on definitions:  df-bi 117  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-ral 2473  df-rex 2474  df-rmo 2476

This theorem is referenced by: (None)