Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > ertrd | Structured version Visualization version GIF version |
Description: A transitivity relation for equivalences. (Contributed by Mario Carneiro, 9-Jul-2014.) |
Ref | Expression |
---|---|
ersymb.1 | ⊢ (𝜑 → 𝑅 Er 𝑋) |
ertrd.5 | ⊢ (𝜑 → 𝐴𝑅𝐵) |
ertrd.6 | ⊢ (𝜑 → 𝐵𝑅𝐶) |
Ref | Expression |
---|---|
ertrd | ⊢ (𝜑 → 𝐴𝑅𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ertrd.5 | . 2 ⊢ (𝜑 → 𝐴𝑅𝐵) | |
2 | ertrd.6 | . 2 ⊢ (𝜑 → 𝐵𝑅𝐶) | |
3 | ersymb.1 | . . 3 ⊢ (𝜑 → 𝑅 Er 𝑋) | |
4 | 3 | ertr 8406 | . 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 5053 Er wer 8388 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-ext 2708 ax-sep 5192 ax-nul 5199 ax-pr 5322 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-sb 2071 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3066 df-rex 3067 df-rab 3070 df-v 3410 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-nul 4238 df-if 4440 df-sn 4542 df-pr 4544 df-op 4548 df-br 5054 df-opab 5116 df-xp 5557 df-rel 5558 df-co 5560 df-er 8391 |
This theorem is referenced by: ertr2d 8408 ertr3d 8409 ertr4d 8410 erinxp 8473 nqereq 10549 adderpq 10570 mulerpq 10571 efgred2 19143 efgcpbllemb 19145 efgcpbl2 19147 pcophtb 23926 pi1xfr 23952 pi1xfrcnvlem 23953 erbr3b 30676 prjspner1 40171 |
Copyright terms: Public domain | W3C validator |