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| Mirrors > Home > MPE Home > Th. List > eqbrtrrdi | Structured version Visualization version GIF version | ||
| Description: A chained equality inference for a binary relation. (Contributed by NM, 4-Jan-2006.) | 
| Ref | Expression | 
|---|---|
| eqbrtrrdi.1 | ⊢ (𝜑 → 𝐵 = 𝐴) | 
| eqbrtrrdi.2 | ⊢ 𝐵𝑅𝐶 | 
| Ref | Expression | 
|---|---|
| eqbrtrrdi | ⊢ (𝜑 → 𝐴𝑅𝐶) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | eqbrtrrdi.1 | . . 3 ⊢ (𝜑 → 𝐵 = 𝐴) | |
| 2 | 1 | eqcomd 2742 | . 2 ⊢ (𝜑 → 𝐴 = 𝐵) | 
| 3 | eqbrtrrdi.2 | . 2 ⊢ 𝐵𝑅𝐶 | |
| 4 | 2, 3 | eqbrtrdi 5181 | 1 ⊢ (𝜑 → 𝐴𝑅𝐶) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 = wceq 1539 class class class wbr 5142 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-ext 2707 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-sb 2064 df-clab 2714 df-cleq 2728 df-clel 2815 df-rab 3436 df-v 3481 df-dif 3953 df-un 3955 df-ss 3967 df-nul 4333 df-if 4525 df-sn 4626 df-pr 4628 df-op 4632 df-br 5143 | 
| This theorem is referenced by: grur1 10861 t1connperf 23445 basellem9 27133 sqff1o 27226 ballotlemic 34510 ballotlem1c 34511 pibt2 37419 | 
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