Theorem breq2dd 32694

Description: Equality deduction for a binary relation. (Contributed by Thierry Arnoux, 10-Jan-2026.)

Hypotheses

Ref	Expression
breq2dd.1	⊢ (𝜑 → 𝐴 = 𝐵)
breq2dd.2	⊢ (𝜑 → 𝐶𝑅𝐴)

Assertion

Ref	Expression
breq2dd	⊢ (𝜑 → 𝐶𝑅𝐵)

Proof of Theorem breq2dd

Step	Hyp	Ref	Expression
1		breq2dd.2	. 2 ⊢ (𝜑 → 𝐶𝑅𝐴)
2		breq2dd.1	. . 3 ⊢ (𝜑 → 𝐴 = 𝐵)
3	2	breq2d 5112	. 2 ⊢ (𝜑 → (𝐶𝑅𝐴 ↔ 𝐶𝑅𝐵))
4	1, 3	mpbid 232	1 ⊢ (𝜑 → 𝐶𝑅𝐵)

Syntax hints:   → wi 4   = wceq 1542   class class class wbr 5100

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-rab 3402  df-v 3444  df-dif 3906  df-un 3908  df-ss 3920  df-nul 4288  df-if 4482  df-sn 4583  df-pr 4585  df-op 4589  df-br 5101

This theorem is referenced by: (None)