Theorem eqbrriv 5215

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 4654	. . 3 |- ( x A y <-> <. x , y >. e. A )
5		df-br 4654	. . 3 |- ( x B y <-> <. x , y >. e. B )
6	3, 4, 5	3bitr3i 290	. 2 |- ( <. x , y >. e. A <-> <. x , y >. e. B )
7	1, 2, 6	eqrelriiv 5214	1 |- A = B

Syntax hints:   <-> wb 196   = wceq 1483   e. wcel 1990   <.cop 4183   class class class wbr 4653   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

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-clab 2609  df-cleq 2615  df-clel 2618  df-in 3581  df-ss 3588  df-br 4654  df-opab 4713  df-xp 5120  df-rel 5121

This theorem is referenced by:  resco  5639  tpostpos  7372  sbthcl  8082  dfle2  11980  dflt2  11981  idsset  31997  dfbigcup2  32006  imageval  32037