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Theorem eqbrb 38213
Description: Substitution of equal classes in a binary relation. (Contributed by Peter Mazsa, 14-Jun-2024.)
Assertion
Ref Expression
eqbrb ((𝐴 = 𝐵𝐴𝑅𝐶) ↔ (𝐴 = 𝐵𝐵𝑅𝐶))

Proof of Theorem eqbrb
StepHypRef Expression
1 simpl 482 . . . 4 ((𝐵 = 𝐴𝐴𝑅𝐶) → 𝐵 = 𝐴)
2 eqbrtr 38212 . . . 4 ((𝐵 = 𝐴𝐴𝑅𝐶) → 𝐵𝑅𝐶)
31, 2jca 511 . . 3 ((𝐵 = 𝐴𝐴𝑅𝐶) → (𝐵 = 𝐴𝐵𝑅𝐶))
4 eqcom 2741 . . . 4 (𝐵 = 𝐴𝐴 = 𝐵)
54anbi1i 624 . . 3 ((𝐵 = 𝐴𝐴𝑅𝐶) ↔ (𝐴 = 𝐵𝐴𝑅𝐶))
64anbi1i 624 . . 3 ((𝐵 = 𝐴𝐵𝑅𝐶) ↔ (𝐴 = 𝐵𝐵𝑅𝐶))
73, 5, 63imtr3i 291 . 2 ((𝐴 = 𝐵𝐴𝑅𝐶) → (𝐴 = 𝐵𝐵𝑅𝐶))
8 simpl 482 . . 3 ((𝐴 = 𝐵𝐵𝑅𝐶) → 𝐴 = 𝐵)
9 eqbrtr 38212 . . 3 ((𝐴 = 𝐵𝐵𝑅𝐶) → 𝐴𝑅𝐶)
108, 9jca 511 . 2 ((𝐴 = 𝐵𝐵𝑅𝐶) → (𝐴 = 𝐵𝐴𝑅𝐶))
117, 10impbii 209 1 ((𝐴 = 𝐵𝐴𝑅𝐶) ↔ (𝐴 = 𝐵𝐵𝑅𝐶))
Colors of variables: wff setvar class
Syntax hints:  wb 206  wa 395   = wceq 1536   class class class wbr 5147
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-ext 2705
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-sb 2062  df-clab 2712  df-cleq 2726  df-clel 2813  df-rab 3433  df-v 3479  df-dif 3965  df-un 3967  df-ss 3979  df-nul 4339  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-br 5148
This theorem is referenced by:  ressn2  38423  trressn  38426
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