Theorem eqbrrdv 4823

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 4089	. . . 4 |- ( x A y <-> <. x , y >. e. A )
3		df-br 4089	. . . 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 1923	. 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 4815	. . 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 1395   = wceq 1397   e. wcel 2202   <.cop 3672   class class class wbr 4088   Rel wrel 4730

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-v 2804  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-br 4089  df-opab 4151  df-xp 4731  df-rel 4732

This theorem is referenced by:  eqbrrdva  4900