Theorem ssrmof 3210

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 3138	. . . 4 |- ( A C_ B <-> A. x ( x e. A -> x e. B ) )
4	3	biimpi 119	. . 3 |- ( A C_ B -> A. x ( x e. A -> x e. B ) )
5		pm3.45 592	. . . 4 |- ( ( x e. A -> x e. B ) -> ( ( x e. A /\ ph ) -> ( x e. B /\ ph ) ) )
6	5	alimi 1448	. . 3 |- ( A. x ( x e. A -> x e. B ) -> A. x ( ( x e. A /\ ph ) -> ( x e. B /\ ph ) ) )
7		moim 2083	. . 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 2456	. 2 |- ( E* x e. B ph <-> E* x ( x e. B /\ ph ) )
10		df-rmo 2456	. 2 |- ( E* x e. A ph <-> E* x ( x e. A /\ ph ) )
11	8, 9, 10	3imtr4g 204	1 |- ( A C_ B -> ( E* x e. B ph -> E* x e. A ph ) )

Syntax hints:   -> wi 4   /\ wa 103   A.wal 1346   E*wmo 2020   e. wcel 2141   F/_wnfc 2299   E*wrmo 2451   C_ wss 3121

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-rmo 2456  df-in 3127  df-ss 3134

This theorem is referenced by: (None)