Theorem eqbrrdv 5217

Description: Deduction from extensionality principle for relations. (Contributed by Mario Carneiro, 3-Jan-2017.)

Hypotheses

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

Assertion

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

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

Proof of Theorem eqbrrdv

Step	Hyp	Ref	Expression
1		eqbrrdv.3	. . . 4 |- ( ph -> ( x A y <-> x B y ) )
2		df-br 4654	. . . 4 |- ( x A y <-> <. x , y >. e. A )
3		df-br 4654	. . . 4 |- ( x B y <-> <. x , y >. e. B )
4	1, 2, 3	3bitr3g 302	. . 3 |- ( ph -> ( <. x , y >. e. A <-> <. x , y >. e. B ) )
5	4	alrimivv 1856	. 2 |- ( ph -> A. x A. y ( <. x , y >. e. A <-> <. x , y >. e. B ) )
6		eqbrrdv.1	. . 3 |- ( ph -> Rel A )
7		eqbrrdv.2	. . 3 |- ( ph -> Rel B )
8		eqrel 5209	. . 3 |- ( ( Rel A /\ Rel B ) -> ( A = B <-> A. x A. y ( <. x , y >. e. A <-> <. x , y >. e. B ) ) )
9	6, 7, 8	syl2anc 693	. 2 |- ( ph -> ( A = B <-> A. x A. y ( <. x , y >. e. A <-> <. x , y >. e. B ) ) )
10	5, 9	mpbird 247	1 |- ( ph -> A = B )

Syntax hints:   -> wi 4   <-> wb 196   A.wal 1481   = 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:  eqbrrdva  5291  oppcsect2  16439