Theorem eqbrb 36438

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

Assertion

Ref	Expression
eqbrb	⊢ ((𝐴 = 𝐵 ∧ 𝐴𝑅𝐶) ↔ (𝐴 = 𝐵 ∧ 𝐵𝑅𝐶))

Proof of Theorem eqbrb

Step	Hyp	Ref	Expression
1		simpl 484	. . . 4 ⊢ ((𝐵 = 𝐴 ∧ 𝐴𝑅𝐶) → 𝐵 = 𝐴)
2		eqbrtr 36437	. . . 4 ⊢ ((𝐵 = 𝐴 ∧ 𝐴𝑅𝐶) → 𝐵𝑅𝐶)
3	1, 2	jca 513	. . 3 ⊢ ((𝐵 = 𝐴 ∧ 𝐴𝑅𝐶) → (𝐵 = 𝐴 ∧ 𝐵𝑅𝐶))
4		eqcom 2743	. . . 4 ⊢ (𝐵 = 𝐴 ↔ 𝐴 = 𝐵)
5	4	anbi1i 625	. . 3 ⊢ ((𝐵 = 𝐴 ∧ 𝐴𝑅𝐶) ↔ (𝐴 = 𝐵 ∧ 𝐴𝑅𝐶))
6	4	anbi1i 625	. . 3 ⊢ ((𝐵 = 𝐴 ∧ 𝐵𝑅𝐶) ↔ (𝐴 = 𝐵 ∧ 𝐵𝑅𝐶))
7	3, 5, 6	3imtr3i 291	. 2 ⊢ ((𝐴 = 𝐵 ∧ 𝐴𝑅𝐶) → (𝐴 = 𝐵 ∧ 𝐵𝑅𝐶))
8		simpl 484	. . 3 ⊢ ((𝐴 = 𝐵 ∧ 𝐵𝑅𝐶) → 𝐴 = 𝐵)
9		eqbrtr 36437	. . 3 ⊢ ((𝐴 = 𝐵 ∧ 𝐵𝑅𝐶) → 𝐴𝑅𝐶)
10	8, 9	jca 513	. 2 ⊢ ((𝐴 = 𝐵 ∧ 𝐵𝑅𝐶) → (𝐴 = 𝐵 ∧ 𝐴𝑅𝐶))
11	7, 10	impbii 208	1 ⊢ ((𝐴 = 𝐵 ∧ 𝐴𝑅𝐶) ↔ (𝐴 = 𝐵 ∧ 𝐵𝑅𝐶))

Syntax hints:   ↔ wb 205   ∧ wa 397   = wceq 1539   class class class wbr 5081

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-ext 2707

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 846  df-3an 1089  df-tru 1542  df-fal 1552  df-ex 1780  df-sb 2066  df-clab 2714  df-cleq 2728  df-clel 2814  df-rab 3333  df-v 3439  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-nul 4263  df-if 4466  df-sn 4566  df-pr 4568  df-op 4572  df-br 5082

This theorem is referenced by:  ressn2  36656  trressn  36659