Theorem eqbrrdv 4816

Description: Deduction from extensionality principle for relations. (Contributed by Mario Carneiro, 3-Jan-2017.)

Hypotheses

Ref	Expression
eqbrrdv.1	|- ( ph -> Rel A )
eqbrrdv.2	|- ( ph -> Rel B )
eqbrrdv.3	|- ( ph -> ( x A y <-> x B y ) )

Assertion

Ref	Expression
eqbrrdv	|- ( ph -> A = B )

Distinct variable groups:   x, y, A   x, B, y   ph, x, y

Proof of Theorem eqbrrdv

Step	Hyp	Ref	Expression
1		eqbrrdv.3	. . . 4 |- ( ph -> ( x A y <-> x B y ) )
2		df-br 4084	. . . 4 |- ( x A y <-> <. x , y >. e. A )
3		df-br 4084	. . . 4 |- ( x B y <-> <. x , y >. e. B )
4	1, 2, 3	3bitr3g 222	. . 3 |- ( ph -> ( <. x , y >. e. A <-> <. x , y >. e. B ) )
5	4	alrimivv 1921	. 2 |- ( ph -> A. x A. y ( <. x , y >. e. A <-> <. x , y >. e. B ) )
6		eqbrrdv.1	. . 3 |- ( ph -> Rel A )
7		eqbrrdv.2	. . 3 |- ( ph -> Rel B )
8		eqrel 4808	. . 3 |- ( ( Rel A /\ Rel B ) -> ( A = B <-> A. x A. y ( <. x , y >. e. A <-> <. x , y >. e. B ) ) )
9	6, 7, 8	syl2anc 411	. 2 |- ( ph -> ( A = B <-> A. x A. y ( <. x , y >. e. A <-> <. x , y >. e. B ) ) )
10	5, 9	mpbird 167	1 |- ( ph -> A = B )

Syntax hints:   -> wi 4   <-> wb 105   A.wal 1393   = wceq 1395   e. wcel 2200   <.cop 3669   class class class wbr 4083   Rel wrel 4724

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-pow 4258  ax-pr 4293

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-v 2801  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-br 4084  df-opab 4146  df-xp 4725  df-rel 4726

This theorem is referenced by:  eqbrrdva  4892