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Mirrors > Home > MPE Home > Th. List > Mathboxes > eceq1i | Structured version Visualization version GIF version |
Description: Equality theorem for 𝐶-coset of 𝐴 and 𝐶-coset of 𝐵, inference version. (Contributed by Peter Mazsa, 11-May-2021.) |
Ref | Expression |
---|---|
eceq1i.1 | ⊢ 𝐴 = 𝐵 |
Ref | Expression |
---|---|
eceq1i | ⊢ [𝐴]𝐶 = [𝐵]𝐶 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eceq1i.1 | . 2 ⊢ 𝐴 = 𝐵 | |
2 | eceq1 8320 | . 2 ⊢ (𝐴 = 𝐵 → [𝐴]𝐶 = [𝐵]𝐶) | |
3 | 1, 2 | ax-mp 5 | 1 ⊢ [𝐴]𝐶 = [𝐵]𝐶 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1536 [cec 8280 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1084 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-rab 3146 df-v 3493 df-dif 3932 df-un 3934 df-in 3936 df-ss 3945 df-nul 4285 df-if 4461 df-sn 4561 df-pr 4563 df-op 4567 df-br 5060 df-opab 5122 df-xp 5554 df-cnv 5556 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-ec 8284 |
This theorem is referenced by: (None) |
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