Theorem ereq2 7750

Description: Equality theorem for equivalence predicate. (Contributed by Mario Carneiro, 12-Aug-2015.)

Assertion

Ref	Expression
ereq2	|- ( A = B -> ( R Er A <-> R Er B ) )

Proof of Theorem ereq2

Step	Hyp	Ref	Expression
1		eqeq2 2633	. . 3 |- ( A = B -> ( dom R = A <-> dom R = B ) )
2	1	3anbi2d 1404	. 2 |- ( A = B -> ( ( Rel R /\ dom R = A /\ ( `' R u. ( R o. R ) ) C_ R ) <-> ( Rel R /\ dom R = B /\ ( `' R u. ( R o. R ) ) C_ R ) ) )
3		df-er 7742	. 2 |- ( R Er A <-> ( Rel R /\ dom R = A /\ ( `' R u. ( R o. R ) ) C_ R ) )
4		df-er 7742	. 2 |- ( R Er B <-> ( Rel R /\ dom R = B /\ ( `' R u. ( R o. R ) ) C_ R ) )
5	2, 3, 4	3bitr4g 303	1 |- ( A = B -> ( R Er A <-> R Er B ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ w3a 1037   = wceq 1483   u. cun 3572   C_ wss 3574   `'ccnv 5113   dom cdm 5114   o. ccom 5118   Rel wrel 5119   Er wer 7739

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-ext 2602

This theorem depends on definitions:  df-bi 197  df-an 386  df-3an 1039  df-ex 1705  df-cleq 2615  df-er 7742

This theorem is referenced by:  iserd  7768  efgval  18130  frgp0  18173  frgpmhm  18178