Theorem eqrelrd2 29426

Description: A version of eqrelrdv2 5219 with explicit non-free declarations. (Contributed by Thierry Arnoux, 28-Aug-2017.)

Hypotheses

Ref	Expression
eqrelrd2.1	|- F/ x ph
eqrelrd2.2	|- F/ y ph
eqrelrd2.3	|- F/_ x A
eqrelrd2.4	|- F/_ y A
eqrelrd2.5	|- F/_ x B
eqrelrd2.6	|- F/_ y B
eqrelrd2.7	|- ( ph -> ( <. x , y >. e. A <-> <. x , y >. e. B ) )

Assertion

Ref	Expression
eqrelrd2	|- ( ( ( Rel A /\ Rel B ) /\ ph ) -> A = B )

Distinct variable group:   x, y

Allowed substitution hints:   ph( x, y)   A( x, y)   B( x, y)

Proof of Theorem eqrelrd2

Step	Hyp	Ref	Expression
1		eqrelrd2.1	. . . 4 |- F/ x ph
2		eqrelrd2.2	. . . . 5 |- F/ y ph
3		eqrelrd2.7	. . . . 5 |- ( ph -> ( <. x , y >. e. A <-> <. x , y >. e. B ) )
4	2, 3	alrimi 2082	. . . 4 |- ( ph -> A. y ( <. x , y >. e. A <-> <. x , y >. e. B ) )
5	1, 4	alrimi 2082	. . 3 |- ( ph -> A. x A. y ( <. x , y >. e. A <-> <. x , y >. e. B ) )
6	5	adantl 482	. 2 |- ( ( ( Rel A /\ Rel B ) /\ ph ) -> A. x A. y ( <. x , y >. e. A <-> <. x , y >. e. B ) )
7		eqrelrd2.3	. . . . . 6 |- F/_ x A
8		eqrelrd2.4	. . . . . 6 |- F/_ y A
9		eqrelrd2.5	. . . . . 6 |- F/_ x B
10		eqrelrd2.6	. . . . . 6 |- F/_ y B
11	1, 2, 7, 8, 9, 10	ssrelf 29425	. . . . 5 |- ( Rel A -> ( A C_ B <-> A. x A. y ( <. x , y >. e. A -> <. x , y >. e. B ) ) )
12	1, 2, 9, 10, 7, 8	ssrelf 29425	. . . . 5 |- ( Rel B -> ( B C_ A <-> A. x A. y ( <. x , y >. e. B -> <. x , y >. e. A ) ) )
13	11, 12	bi2anan9 917	. . . 4 |- ( ( Rel A /\ Rel B ) -> ( ( A C_ B /\ B C_ A ) <-> ( A. x A. y ( <. x , y >. e. A -> <. x , y >. e. B ) /\ A. x A. y ( <. x , y >. e. B -> <. x , y >. e. A ) ) ) )
14		eqss 3618	. . . 4 |- ( A = B <-> ( A C_ B /\ B C_ A ) )
15		2albiim 1817	. . . 4 |- ( A. x A. y ( <. x , y >. e. A <-> <. x , y >. e. B ) <-> ( A. x A. y ( <. x , y >. e. A -> <. x , y >. e. B ) /\ A. x A. y ( <. x , y >. e. B -> <. x , y >. e. A ) ) )
16	13, 14, 15	3bitr4g 303	. . 3 |- ( ( Rel A /\ Rel B ) -> ( A = B <-> A. x A. y ( <. x , y >. e. A <-> <. x , y >. e. B ) ) )
17	16	adantr 481	. 2 |- ( ( ( Rel A /\ Rel B ) /\ ph ) -> ( A = B <-> A. x A. y ( <. x , y >. e. A <-> <. x , y >. e. B ) ) )
18	6, 17	mpbird 247	1 |- ( ( ( Rel A /\ Rel B ) /\ ph ) -> A = B )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   A.wal 1481   = wceq 1483   F/wnf 1708   e. wcel 1990   F/_wnfc 2751   C_ wss 3574   <.cop 4183   Rel wrel 5119

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  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-opab 4713  df-xp 5120  df-rel 5121

This theorem is referenced by:  fpwrelmap  29508