Theorem eqbrrdiv 4726

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 4006	. . 3 |- ( x A y <-> <. x , y >. e. A )
5		df-br 4006	. . 3 |- ( x B y <-> <. x , y >. e. B )
6	3, 4, 5	3bitr3g 222	. 2 |- ( ph -> ( <. x , y >. e. A <-> <. x , y >. e. B ) )
7	1, 2, 6	eqrelrdv 4724	1 |- ( ph -> A = B )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1353   e. wcel 2148   <.cop 3597   class class class wbr 4005   Rel wrel 4633

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-pow 4176  ax-pr 4211

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2741  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-br 4006  df-opab 4067  df-xp 4634  df-rel 4635

This theorem is referenced by: (None)