Theorem reu2eqd 3403

Description: Deduce equality from restricted uniqueness, deduction version. (Contributed by Thierry Arnoux, 27-Nov-2019.)

Hypotheses

Ref	Expression
reu2eqd.1	|- ( x = B -> ( ps <-> ch ) )
reu2eqd.2	|- ( x = C -> ( ps <-> th ) )
reu2eqd.3	|- ( ph -> E! x e. A ps )
reu2eqd.4	|- ( ph -> B e. A )
reu2eqd.5	|- ( ph -> C e. A )
reu2eqd.6	|- ( ph -> ch )
reu2eqd.7	|- ( ph -> th )

Assertion

Ref	Expression
reu2eqd	|- ( ph -> B = C )

Distinct variable groups:   x, A   x, B   x, C   ch, x   th, x

Allowed substitution hints:   ph( x)   ps( x)

Proof of Theorem reu2eqd

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		reu2eqd.6	. 2 |- ( ph -> ch )
2		reu2eqd.7	. 2 |- ( ph -> th )
3		reu2eqd.3	. . . . 5 |- ( ph -> E! x e. A ps )
4		reu2 3394	. . . . 5 |- ( E! x e. A ps <-> ( E. x e. A ps /\ A. x e. A A. y e. A ( ( ps /\ [ y / x ] ps ) -> x = y ) ) )
5	3, 4	sylib 208	. . . 4 |- ( ph -> ( E. x e. A ps /\ A. x e. A A. y e. A ( ( ps /\ [ y / x ] ps ) -> x = y ) ) )
6	5	simprd 479	. . 3 |- ( ph -> A. x e. A A. y e. A ( ( ps /\ [ y / x ] ps ) -> x = y ) )
7		reu2eqd.4	. . . 4 |- ( ph -> B e. A )
8		reu2eqd.5	. . . 4 |- ( ph -> C e. A )
9		nfv 1843	. . . . . . 7 |- F/ x ch
10		nfs1v 2437	. . . . . . 7 |- F/ x [ y / x ] ps
11	9, 10	nfan 1828	. . . . . 6 |- F/ x ( ch /\ [ y / x ] ps )
12		nfv 1843	. . . . . 6 |- F/ x B = y
13	11, 12	nfim 1825	. . . . 5 |- F/ x ( ( ch /\ [ y / x ] ps ) -> B = y )
14		nfv 1843	. . . . 5 |- F/ y ( ( ch /\ th ) -> B = C )
15		reu2eqd.1	. . . . . . 7 |- ( x = B -> ( ps <-> ch ) )
16	15	anbi1d 741	. . . . . 6 |- ( x = B -> ( ( ps /\ [ y / x ] ps ) <-> ( ch /\ [ y / x ] ps ) ) )
17		eqeq1 2626	. . . . . 6 |- ( x = B -> ( x = y <-> B = y ) )
18	16, 17	imbi12d 334	. . . . 5 |- ( x = B -> ( ( ( ps /\ [ y / x ] ps ) -> x = y ) <-> ( ( ch /\ [ y / x ] ps ) -> B = y ) ) )
19		nfv 1843	. . . . . . . 8 |- F/ x th
20		reu2eqd.2	. . . . . . . 8 |- ( x = C -> ( ps <-> th ) )
21	19, 20	sbhypf 3253	. . . . . . 7 |- ( y = C -> ( [ y / x ] ps <-> th ) )
22	21	anbi2d 740	. . . . . 6 |- ( y = C -> ( ( ch /\ [ y / x ] ps ) <-> ( ch /\ th ) ) )
23		eqeq2 2633	. . . . . 6 |- ( y = C -> ( B = y <-> B = C ) )
24	22, 23	imbi12d 334	. . . . 5 |- ( y = C -> ( ( ( ch /\ [ y / x ] ps ) -> B = y ) <-> ( ( ch /\ th ) -> B = C ) ) )
25	13, 14, 18, 24	rspc2 3320	. . . 4 |- ( ( B e. A /\ C e. A ) -> ( A. x e. A A. y e. A ( ( ps /\ [ y / x ] ps ) -> x = y ) -> ( ( ch /\ th ) -> B = C ) ) )
26	7, 8, 25	syl2anc 693	. . 3 |- ( ph -> ( A. x e. A A. y e. A ( ( ps /\ [ y / x ] ps ) -> x = y ) -> ( ( ch /\ th ) -> B = C ) ) )
27	6, 26	mpd 15	. 2 |- ( ph -> ( ( ch /\ th ) -> B = C ) )
28	1, 2, 27	mp2and 715	1 |- ( ph -> B = C )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   [wsb 1880   e. wcel 1990   A.wral 2912   E.wrex 2913   E!wreu 2914

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-ral 2917  df-rex 2918  df-reu 2919  df-v 3202

This theorem is referenced by:  qtophmeo  21620  footeq  25616  mideulem2  25626  lmieq  25683