Theorem eqbrrdiv 4597

Description: Deduction from extensionality principle for relations. (Contributed by Rodolfo Medina, 10-Oct-2010.)

Hypotheses

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

Assertion

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

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

Proof of Theorem eqbrrdiv

Step	Hyp	Ref	Expression
1		eqbrrdiv.1	. 2 |- Rel A
2		eqbrrdiv.2	. 2 |- Rel B
3		eqbrrdiv.3	. . 3 |- ( ph -> ( x A y <-> x B y ) )
4		df-br 3896	. . 3 |- ( x A y <-> <. x , y >. e. A )
5		df-br 3896	. . 3 |- ( x B y <-> <. x , y >. e. B )
6	3, 4, 5	3bitr3g 221	. 2 |- ( ph -> ( <. x , y >. e. A <-> <. x , y >. e. B ) )
7	1, 2, 6	eqrelrdv 4595	1 |- ( ph -> A = B )

Syntax hints:   -> wi 4   <-> wb 104   = wceq 1314   e. wcel 1463   <.cop 3496   class class class wbr 3895   Rel wrel 4504

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4006  ax-pow 4058  ax-pr 4091

This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2244  df-v 2659  df-un 3041  df-in 3043  df-ss 3050  df-pw 3478  df-sn 3499  df-pr 3500  df-op 3502  df-br 3896  df-opab 3950  df-xp 4505  df-rel 4506

This theorem is referenced by: (None)