Theorem eqbrriv 4821

Description: Inference from extensionality principle for relations. (Contributed by NM, 12-Dec-2006.)

Hypotheses

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

Assertion

Ref	Expression
eqbrriv	|- A = B

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

Proof of Theorem eqbrriv

Step	Hyp	Ref	Expression
1		eqbrriv.1	. 2 |- Rel A
2		eqbrriv.2	. 2 |- Rel B
3		eqbrriv.3	. . 3 |- ( x A y <-> x B y )
4		df-br 4089	. . 3 |- ( x A y <-> <. x , y >. e. A )
5		df-br 4089	. . 3 |- ( x B y <-> <. x , y >. e. B )
6	3, 4, 5	3bitr3i 210	. 2 |- ( <. x , y >. e. A <-> <. x , y >. e. B )
7	1, 2, 6	eqrelriiv 4820	1 |- A = B

Syntax hints:   <-> wb 105   = 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:  resco  5241  tpostpos  6429