Theorem ssrmo 29334

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

Hypotheses

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

Assertion

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

Proof of Theorem ssrmo

Step	Hyp	Ref	Expression
1		ssrmo.1	. . . . 5 |- F/_ x A
2		ssrmo.2	. . . . 5 |- F/_ x B
3	1, 2	dfss2f 3594	. . . 4 |- ( A C_ B <-> A. x ( x e. A -> x e. B ) )
4	3	biimpi 206	. . 3 |- ( A C_ B -> A. x ( x e. A -> x e. B ) )
5		pm3.45 879	. . . 4 |- ( ( x e. A -> x e. B ) -> ( ( x e. A /\ ph ) -> ( x e. B /\ ph ) ) )
6	5	alimi 1739	. . 3 |- ( A. x ( x e. A -> x e. B ) -> A. x ( ( x e. A /\ ph ) -> ( x e. B /\ ph ) ) )
7		moim 2519	. . 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 18	. 2 |- ( A C_ B -> ( E* x ( x e. B /\ ph ) -> E* x ( x e. A /\ ph ) ) )
9		df-rmo 2920	. 2 |- ( E* x e. B ph <-> E* x ( x e. B /\ ph ) )
10		df-rmo 2920	. 2 |- ( E* x e. A ph <-> E* x ( x e. A /\ ph ) )
11	8, 9, 10	3imtr4g 285	1 |- ( A C_ B -> ( E* x e. B ph -> E* x e. A ph ) )

Syntax hints:   -> wi 4   /\ wa 384   A.wal 1481   e. wcel 1990   E*wmo 2471   F/_wnfc 2751   E*wrmo 2915   C_ wss 3574

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-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-rmo 2920  df-in 3581  df-ss 3588

This theorem is referenced by:  disjss1f  29386