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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 8740 | . 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 1533 [cec 8698 |
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 2695 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-sb 2060 df-clab 2702 df-cleq 2716 df-clel 2802 df-rab 3425 df-v 3468 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-nul 4316 df-if 4522 df-sn 4622 df-pr 4624 df-op 4628 df-br 5140 df-opab 5202 df-cnv 5675 df-dm 5677 df-rn 5678 df-res 5679 df-ima 5680 df-ec 8702 |
This theorem is referenced by: vrgpfval 19682 quslsm 33012 opprqusplusg 33099 opprqusmulr 33101 qsdrngi 33105 releldmqscoss 38034 prjspeclsp 41906 |
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