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Mirrors > Home > MPE Home > Th. List > eceq2d | Structured version Visualization version GIF version |
Description: Equality theorem for the 𝐴-coset and 𝐵-coset of 𝐶, deduction version. (Contributed by Peter Mazsa, 23-Apr-2021.) |
Ref | Expression |
---|---|
eceq2d.1 | ⊢ (𝜑 → 𝐴 = 𝐵) |
Ref | Expression |
---|---|
eceq2d | ⊢ (𝜑 → [𝐶]𝐴 = [𝐶]𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eceq2d.1 | . 2 ⊢ (𝜑 → 𝐴 = 𝐵) | |
2 | eceq2 8513 | . 2 ⊢ (𝐴 = 𝐵 → [𝐶]𝐴 = [𝐶]𝐵) | |
3 | 1, 2 | syl 17 | 1 ⊢ (𝜑 → [𝐶]𝐴 = [𝐶]𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 [cec 8471 |
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 2015 ax-8 2112 ax-9 2120 ax-ext 2711 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1545 df-fal 1555 df-ex 1787 df-sb 2072 df-clab 2718 df-cleq 2732 df-clel 2818 df-rab 3075 df-v 3433 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-nul 4263 df-if 4466 df-sn 4568 df-pr 4570 df-op 4574 df-br 5080 df-opab 5142 df-cnv 5597 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-ec 8475 |
This theorem is referenced by: vrgpfval 19362 quslsm 31581 releldmqscoss 36760 prjspeclsp 40440 |
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