Theorem breqdi 3997

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 3993	. 2 |- ( ph -> ( C A D <-> C B D ) )
4	1, 3	mpbid 146	1 |- ( ph -> C B D )

Syntax hints:   -> wi 4   = wceq 1343   class class class wbr 3982

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 1435  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-4 1498  ax-17 1514  ax-ial 1522  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-cleq 2158  df-clel 2161  df-br 3983

This theorem is referenced by:  dvef  13328