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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 8329 | . 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 1537 [cec 8287 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-rab 3147 df-v 3496 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-op 4574 df-br 5067 df-opab 5129 df-cnv 5563 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-ec 8291 |
This theorem is referenced by: vrgpfval 18892 releldmqscoss 35909 prjspeclsp 39311 |
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