Theorem reuimrmo 41178

Description: Restricted uniqueness implies restricted "at most one" through implication, analogous to euimmo 2522. (Contributed by Alexander van der Vekens, 25-Jun-2017.)

Assertion

Ref	Expression
reuimrmo	|- ( A. x e. A ( ph -> ps ) -> ( E! x e. A ps -> E* x e. A ph ) )

Proof of Theorem reuimrmo

Step	Hyp	Ref	Expression
1		reurmo 3161	. 2 |- ( E! x e. A ps -> E* x e. A ps )
2		rmoim 3407	. 2 |- ( A. x e. A ( ph -> ps ) -> ( E* x e. A ps -> E* x e. A ph ) )
3	1, 2	syl5 34	1 |- ( A. x e. A ( ph -> ps ) -> ( E! x e. A ps -> E* x e. A ph ) )

Syntax hints:   -> wi 4   A.wral 2912   E!wreu 2914   E*wrmo 2915

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-10 2019  ax-12 2047

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-ex 1705  df-nf 1710  df-eu 2474  df-mo 2475  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920

This theorem is referenced by:  2reurmo  41182