Theorem eqrel2 34068

Description: Equality of relations. (Contributed by Peter Mazsa, 8-Mar-2019.)

Assertion

Ref	Expression
eqrel2	|- ( ( Rel A /\ Rel B ) -> ( A = B <-> A. x A. y ( x A y <-> x B y ) ) )

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

Proof of Theorem eqrel2

Step	Hyp	Ref	Expression
1		ssrel3 34067	. . 3 |- ( Rel A -> ( A C_ B <-> A. x A. y ( x A y -> x B y ) ) )
2		ssrel3 34067	. . 3 |- ( Rel B -> ( B C_ A <-> A. x A. y ( x B y -> x A y ) ) )
3	1, 2	bi2anan9 917	. 2 |- ( ( Rel A /\ Rel B ) -> ( ( A C_ B /\ B C_ A ) <-> ( A. x A. y ( x A y -> x B y ) /\ A. x A. y ( x B y -> x A y ) ) ) )
4		eqss 3618	. 2 |- ( A = B <-> ( A C_ B /\ B C_ A ) )
5		2albiim 1817	. 2 |- ( A. x A. y ( x A y <-> x B y ) <-> ( A. x A. y ( x A y -> x B y ) /\ A. x A. y ( x B y -> x A y ) ) )
6	3, 4, 5	3bitr4g 303	1 |- ( ( Rel A /\ Rel B ) -> ( A = B <-> A. x A. y ( x A y <-> x B y ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   A.wal 1481   = wceq 1483   C_ wss 3574   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: (None)