Theorem eqbrtr 38441

Description: Substitution of equal classes in binary relation. (Contributed by Peter Mazsa, 14-Jun-2024.)

Assertion

Ref	Expression
eqbrtr	⊢ ((𝐴 = 𝐵 ∧ 𝐵𝑅𝐶) → 𝐴𝑅𝐶)

Proof of Theorem eqbrtr

Step	Hyp	Ref	Expression
1		breq1 5102	. 2 ⊢ (𝐴 = 𝐵 → (𝐴𝑅𝐶 ↔ 𝐵𝑅𝐶))
2	1	biimpar 477	1 ⊢ ((𝐴 = 𝐵 ∧ 𝐵𝑅𝐶) → 𝐴𝑅𝐶)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   class class class wbr 5099

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 3401  df-v 3443  df-dif 3905  df-un 3907  df-ss 3919  df-nul 4287  df-if 4481  df-sn 4582  df-pr 4584  df-op 4588  df-br 5100

This theorem is referenced by:  eqbrb  38442