Theorem eqbrrdiv 4702

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 3983	. . 3 |- ( x A y <-> <. x , y >. e. A )
5		df-br 3983	. . 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 4700	1 |- ( ph -> A = B )

Syntax hints:   -> wi 4   <-> wb 104   = wceq 1343   e. wcel 2136   <.cop 3579   class class class wbr 3982   Rel wrel 4609

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-v 2728  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-br 3983  df-opab 4044  df-xp 4610  df-rel 4611

This theorem is referenced by: (None)