Theorem breqdi 4033

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 4029	. 2 |- ( ph -> ( C A D <-> C B D ) )
4	1, 3	mpbid 147	1 |- ( ph -> C B D )

Syntax hints:   -> wi 4   = wceq 1364   class class class wbr 4018

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-5 1458  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-4 1521  ax-17 1537  ax-ial 1545  ax-ext 2171

This theorem depends on definitions:  df-bi 117  df-cleq 2182  df-clel 2185  df-br 4019

This theorem is referenced by:  dvef  14640