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Mirrors > Home > MPE Home > Th. List > Mathboxes > eqvreltr4d | 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 |
---|---|
eqvreltr4d.1 | ⊢ (𝜑 → EqvRel 𝑅) |
eqvreltr4d.2 | ⊢ (𝜑 → 𝐴𝑅𝐵) |
eqvreltr4d.3 | ⊢ (𝜑 → 𝐶𝑅𝐵) |
Ref | Expression |
---|---|
eqvreltr4d | ⊢ (𝜑 → 𝐴𝑅𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqvreltr4d.1 | . 2 ⊢ (𝜑 → EqvRel 𝑅) | |
2 | eqvreltr4d.2 | . 2 ⊢ (𝜑 → 𝐴𝑅𝐵) | |
3 | eqvreltr4d.3 | . . 3 ⊢ (𝜑 → 𝐶𝑅𝐵) | |
4 | 1, 3 | eqvrelsym 38133 | . 2 ⊢ (𝜑 → 𝐵𝑅𝐶) |
5 | 1, 2, 4 | eqvreltrd 38136 | 1 ⊢ (𝜑 → 𝐴𝑅𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 class class class wbr 5143 EqvRel weqvrel 37722 |
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 2696 ax-sep 5294 ax-nul 5301 ax-pr 5423 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-sb 2060 df-clab 2703 df-cleq 2717 df-clel 2802 df-ral 3052 df-rex 3061 df-rab 3420 df-v 3465 df-dif 3942 df-un 3944 df-in 3946 df-ss 3956 df-nul 4319 df-if 4525 df-sn 4625 df-pr 4627 df-op 4631 df-br 5144 df-opab 5206 df-id 5570 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-refrel 38040 df-symrel 38072 df-trrel 38102 df-eqvrel 38113 |
This theorem is referenced by: eqvrelref 38138 eqvreldisj 38142 |
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