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Mirrors > Home > MPE Home > Th. List > Mathboxes > eqvreltrd | Structured version Visualization version GIF version |
Description: A transitivity relation for equivalences. (Contributed by Mario Carneiro, 9-Jul-2014.) (Revised by Peter Mazsa, 2-Jun-2019.) |
Ref | Expression |
---|---|
eqvreltrd.1 | ⊢ (𝜑 → EqvRel 𝑅) |
eqvreltrd.2 | ⊢ (𝜑 → 𝐴𝑅𝐵) |
eqvreltrd.3 | ⊢ (𝜑 → 𝐵𝑅𝐶) |
Ref | Expression |
---|---|
eqvreltrd | ⊢ (𝜑 → 𝐴𝑅𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqvreltrd.2 | . 2 ⊢ (𝜑 → 𝐴𝑅𝐵) | |
2 | eqvreltrd.3 | . 2 ⊢ (𝜑 → 𝐵𝑅𝐶) | |
3 | eqvreltrd.1 | . . 3 ⊢ (𝜑 → EqvRel 𝑅) | |
4 | 3 | eqvreltr 36343 | . 2 ⊢ (𝜑 → ((𝐴𝑅𝐵 ∧ 𝐵𝑅𝐶) → 𝐴𝑅𝐶)) |
5 | 1, 2, 4 | mp2and 699 | 1 ⊢ (𝜑 → 𝐴𝑅𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 class class class wbr 5030 EqvRel weqvrel 35973 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-12 2179 ax-ext 2710 ax-sep 5167 ax-nul 5174 ax-pr 5296 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-clab 2717 df-cleq 2730 df-clel 2811 df-ral 3058 df-rex 3059 df-rab 3062 df-v 3400 df-dif 3846 df-un 3848 df-in 3850 df-ss 3860 df-nul 4212 df-if 4415 df-sn 4517 df-pr 4519 df-op 4523 df-br 5031 df-opab 5093 df-id 5429 df-xp 5531 df-rel 5532 df-cnv 5533 df-co 5534 df-dm 5535 df-rn 5536 df-res 5537 df-refrel 36253 df-symrel 36281 df-trrel 36311 df-eqvrel 36321 |
This theorem is referenced by: eqvreltr4d 36345 |
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