Theorem breqdi 4004

Description: Equality deduction for a binary relation. (Contributed by Thierry Arnoux, 5-Oct-2020.)

Hypotheses

Ref	Expression
breq1d.1	|- ( ph -> A = B )
breqdi.1	|- ( ph -> C A D )

Assertion

Ref	Expression
breqdi	|- ( ph -> C B D )

Proof of Theorem breqdi

Step	Hyp	Ref	Expression
1		breqdi.1	. 2 |- ( ph -> C A D )
2		breq1d.1	. . 3 |- ( ph -> A = B )
3	2	breqd 4000	. 2 |- ( ph -> ( C A D <-> C B D ) )
4	1, 3	mpbid 146	1 |- ( ph -> C B D )

Syntax hints:   -> wi 4   = wceq 1348   class class class wbr 3989

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1440  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-4 1503  ax-17 1519  ax-ial 1527  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-cleq 2163  df-clel 2166  df-br 3990

This theorem is referenced by:  dvef  13482