| Mathbox for Peter Mazsa |
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| Mirrors > Home > MPE Home > Th. List > Mathboxes > eqbrb | Structured version Visualization version GIF version | ||
| Description: Substitution of equal classes in a binary relation. (Contributed by Peter Mazsa, 14-Jun-2024.) |
| Ref | Expression |
|---|---|
| eqbrb | ⊢ ((𝐴 = 𝐵 ∧ 𝐴𝑅𝐶) ↔ (𝐴 = 𝐵 ∧ 𝐵𝑅𝐶)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | simpl 482 | . . . 4 ⊢ ((𝐵 = 𝐴 ∧ 𝐴𝑅𝐶) → 𝐵 = 𝐴) | |
| 2 | eqbrtr 38233 | . . . 4 ⊢ ((𝐵 = 𝐴 ∧ 𝐴𝑅𝐶) → 𝐵𝑅𝐶) | |
| 3 | 1, 2 | jca 511 | . . 3 ⊢ ((𝐵 = 𝐴 ∧ 𝐴𝑅𝐶) → (𝐵 = 𝐴 ∧ 𝐵𝑅𝐶)) |
| 4 | eqcom 2744 | . . . 4 ⊢ (𝐵 = 𝐴 ↔ 𝐴 = 𝐵) | |
| 5 | 4 | anbi1i 624 | . . 3 ⊢ ((𝐵 = 𝐴 ∧ 𝐴𝑅𝐶) ↔ (𝐴 = 𝐵 ∧ 𝐴𝑅𝐶)) |
| 6 | 4 | anbi1i 624 | . . 3 ⊢ ((𝐵 = 𝐴 ∧ 𝐵𝑅𝐶) ↔ (𝐴 = 𝐵 ∧ 𝐵𝑅𝐶)) |
| 7 | 3, 5, 6 | 3imtr3i 291 | . 2 ⊢ ((𝐴 = 𝐵 ∧ 𝐴𝑅𝐶) → (𝐴 = 𝐵 ∧ 𝐵𝑅𝐶)) |
| 8 | simpl 482 | . . 3 ⊢ ((𝐴 = 𝐵 ∧ 𝐵𝑅𝐶) → 𝐴 = 𝐵) | |
| 9 | eqbrtr 38233 | . . 3 ⊢ ((𝐴 = 𝐵 ∧ 𝐵𝑅𝐶) → 𝐴𝑅𝐶) | |
| 10 | 8, 9 | jca 511 | . 2 ⊢ ((𝐴 = 𝐵 ∧ 𝐵𝑅𝐶) → (𝐴 = 𝐵 ∧ 𝐴𝑅𝐶)) |
| 11 | 7, 10 | impbii 209 | 1 ⊢ ((𝐴 = 𝐵 ∧ 𝐴𝑅𝐶) ↔ (𝐴 = 𝐵 ∧ 𝐵𝑅𝐶)) |
| Colors of variables: wff setvar class |
| Syntax hints: ↔ wb 206 ∧ wa 395 = wceq 1540 class class class wbr 5143 |
| 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 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-br 5144 |
| This theorem is referenced by: ressn2 38443 trressn 38446 |
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