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Mirrors > Home > MPE Home > Th. List > eceq2i | Structured version Visualization version GIF version |
Description: Equality theorem for the 𝐴-coset and 𝐵-coset of 𝐶, inference version. (Contributed by Peter Mazsa, 11-May-2021.) |
Ref | Expression |
---|---|
eceq2i.1 | ⊢ 𝐴 = 𝐵 |
Ref | Expression |
---|---|
eceq2i | ⊢ [𝐶]𝐴 = [𝐶]𝐵 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eceq2i.1 | . 2 ⊢ 𝐴 = 𝐵 | |
2 | eceq2 8689 | . 2 ⊢ (𝐴 = 𝐵 → [𝐶]𝐴 = [𝐶]𝐵) | |
3 | 1, 2 | ax-mp 5 | 1 ⊢ [𝐶]𝐴 = [𝐶]𝐵 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1542 [cec 8647 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2708 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-clab 2715 df-cleq 2729 df-clel 2815 df-rab 3409 df-v 3448 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-sn 4588 df-pr 4590 df-op 4594 df-br 5107 df-opab 5169 df-cnv 5642 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-ec 8651 |
This theorem is referenced by: eccnvepres3 36749 extid 36774 br2coss 36903 eldisjlem19 37275 prjspeclsp 40953 |
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