Theorem eqbrrdv 4756

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 4030	. . . 4 |- ( x A y <-> <. x , y >. e. A )
3		df-br 4030	. . . 4 |- ( x B y <-> <. x , y >. e. B )
4	1, 2, 3	3bitr3g 222	. . 3 |- ( ph -> ( <. x , y >. e. A <-> <. x , y >. e. B ) )
5	4	alrimivv 1886	. 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 4748	. . 3 |- ( ( Rel A /\ Rel B ) -> ( A = B <-> A. x A. y ( <. x , y >. e. A <-> <. x , y >. e. B ) ) )
9	6, 7, 8	syl2anc 411	. 2 |- ( ph -> ( A = B <-> A. x A. y ( <. x , y >. e. A <-> <. x , y >. e. B ) ) )
10	5, 9	mpbird 167	1 |- ( ph -> A = B )

Syntax hints:   -> wi 4   <-> wb 105   A.wal 1362   = wceq 1364   e. wcel 2164   <.cop 3621   class class class wbr 4029   Rel wrel 4664

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-pow 4203  ax-pr 4238

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-v 2762  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-br 4030  df-opab 4091  df-xp 4665  df-rel 4666

This theorem is referenced by:  eqbrrdva  4832