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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 37601 | . . . 4 ⊢ ((𝐵 = 𝐴 ∧ 𝐴𝑅𝐶) → 𝐵𝑅𝐶) | |
3 | 1, 2 | jca 511 | . . 3 ⊢ ((𝐵 = 𝐴 ∧ 𝐴𝑅𝐶) → (𝐵 = 𝐴 ∧ 𝐵𝑅𝐶)) |
4 | eqcom 2731 | . . . 4 ⊢ (𝐵 = 𝐴 ↔ 𝐴 = 𝐵) | |
5 | 4 | anbi1i 623 | . . 3 ⊢ ((𝐵 = 𝐴 ∧ 𝐴𝑅𝐶) ↔ (𝐴 = 𝐵 ∧ 𝐴𝑅𝐶)) |
6 | 4 | anbi1i 623 | . . 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 37601 | . . 3 ⊢ ((𝐴 = 𝐵 ∧ 𝐵𝑅𝐶) → 𝐴𝑅𝐶) | |
10 | 8, 9 | jca 511 | . 2 ⊢ ((𝐴 = 𝐵 ∧ 𝐵𝑅𝐶) → (𝐴 = 𝐵 ∧ 𝐴𝑅𝐶)) |
11 | 7, 10 | impbii 208 | 1 ⊢ ((𝐴 = 𝐵 ∧ 𝐴𝑅𝐶) ↔ (𝐴 = 𝐵 ∧ 𝐵𝑅𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ wa 395 = wceq 1533 class class class wbr 5139 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2695 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-sb 2060 df-clab 2702 df-cleq 2716 df-clel 2802 df-rab 3425 df-v 3468 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-nul 4316 df-if 4522 df-sn 4622 df-pr 4624 df-op 4628 df-br 5140 |
This theorem is referenced by: ressn2 37815 trressn 37818 |
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