Theorem eqbrrdiv 4736

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 4016	. . 3 |- ( x A y <-> <. x , y >. e. A )
5		df-br 4016	. . 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 4734	1 |- ( ph -> A = B )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1363   e. wcel 2158   <.cop 3607   class class class wbr 4015   Rel wrel 4643

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 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-14 2161  ax-ext 2169  ax-sep 4133  ax-pow 4186  ax-pr 4221

This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-nf 1471  df-sb 1773  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-v 2751  df-un 3145  df-in 3147  df-ss 3154  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-br 4016  df-opab 4077  df-xp 4644  df-rel 4645

This theorem is referenced by: (None)