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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 2739 | . 2 ⊢ (𝜑 → 𝐴 = 𝐵) |
3 | eqbrtrrdi.2 | . 2 ⊢ 𝐵𝑅𝐶 | |
4 | 2, 3 | eqbrtrdi 5186 | 1 ⊢ (𝜑 → 𝐴𝑅𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 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 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2704 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-rab 3434 df-v 3477 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-sn 4628 df-pr 4630 df-op 4634 df-br 5148 |
This theorem is referenced by: grur1 10811 t1connperf 22922 basellem9 26573 sqff1o 26666 ballotlemic 33443 ballotlem1c 33444 pibt2 36236 |
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