Theorem ssrmof 3255

Description: "At most one" existential quantification restricted to a subclass. (Contributed by Thierry Arnoux, 8-Oct-2017.)

Hypotheses

Ref	Expression
ssrexf.1	|- F/_ x A
ssrexf.2	|- F/_ x B

Assertion

Ref	Expression
ssrmof	|- ( A C_ B -> ( E* x e. B ph -> E* x e. A ph ) )

Proof of Theorem ssrmof

Step	Hyp	Ref	Expression
1		ssrexf.1	. . . . 5 |- F/_ x A
2		ssrexf.2	. . . . 5 |- F/_ x B
3	1, 2	dfss2f 3183	. . . 4 |- ( A C_ B <-> A. x ( x e. A -> x e. B ) )
4	3	biimpi 120	. . 3 |- ( A C_ B -> A. x ( x e. A -> x e. B ) )
5		pm3.45 597	. . . 4 |- ( ( x e. A -> x e. B ) -> ( ( x e. A /\ ph ) -> ( x e. B /\ ph ) ) )
6	5	alimi 1477	. . 3 |- ( A. x ( x e. A -> x e. B ) -> A. x ( ( x e. A /\ ph ) -> ( x e. B /\ ph ) ) )
7		moim 2117	. . 3 |- ( A. x ( ( x e. A /\ ph ) -> ( x e. B /\ ph ) ) -> ( E* x ( x e. B /\ ph ) -> E* x ( x e. A /\ ph ) ) )
8	4, 6, 7	3syl 17	. 2 |- ( A C_ B -> ( E* x ( x e. B /\ ph ) -> E* x ( x e. A /\ ph ) ) )
9		df-rmo 2491	. 2 |- ( E* x e. B ph <-> E* x ( x e. B /\ ph ) )
10		df-rmo 2491	. 2 |- ( E* x e. A ph <-> E* x ( x e. A /\ ph ) )
11	8, 9, 10	3imtr4g 205	1 |- ( A C_ B -> ( E* x e. B ph -> E* x e. A ph ) )

Syntax hints:   -> wi 4   /\ wa 104   A.wal 1370   E*wmo 2054   e. wcel 2175   F/_wnfc 2334   E*wrmo 2486   C_ wss 3165

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 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-ext 2186

This theorem depends on definitions:  df-bi 117  df-nf 1483  df-sb 1785  df-eu 2056  df-mo 2057  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-rmo 2491  df-in 3171  df-ss 3178

This theorem is referenced by: (None)