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Mirrors > Home > MPE Home > Th. List > caovcanrd | Structured version Visualization version GIF version |
Description: Commute the arguments of an operation cancellation law. (Contributed by Mario Carneiro, 30-Dec-2014.) |
Ref | Expression |
---|---|
caovcang.1 | ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑇 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → ((𝑥𝐹𝑦) = (𝑥𝐹𝑧) ↔ 𝑦 = 𝑧)) |
caovcand.2 | ⊢ (𝜑 → 𝐴 ∈ 𝑇) |
caovcand.3 | ⊢ (𝜑 → 𝐵 ∈ 𝑆) |
caovcand.4 | ⊢ (𝜑 → 𝐶 ∈ 𝑆) |
caovcanrd.5 | ⊢ (𝜑 → 𝐴 ∈ 𝑆) |
caovcanrd.6 | ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥𝐹𝑦) = (𝑦𝐹𝑥)) |
Ref | Expression |
---|---|
caovcanrd | ⊢ (𝜑 → ((𝐵𝐹𝐴) = (𝐶𝐹𝐴) ↔ 𝐵 = 𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | caovcanrd.6 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥𝐹𝑦) = (𝑦𝐹𝑥)) | |
2 | caovcanrd.5 | . . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑆) | |
3 | caovcand.3 | . . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝑆) | |
4 | 1, 2, 3 | caovcomd 7344 | . . 3 ⊢ (𝜑 → (𝐴𝐹𝐵) = (𝐵𝐹𝐴)) |
5 | caovcand.4 | . . . 4 ⊢ (𝜑 → 𝐶 ∈ 𝑆) | |
6 | 1, 2, 5 | caovcomd 7344 | . . 3 ⊢ (𝜑 → (𝐴𝐹𝐶) = (𝐶𝐹𝐴)) |
7 | 4, 6 | eqeq12d 2837 | . 2 ⊢ (𝜑 → ((𝐴𝐹𝐵) = (𝐴𝐹𝐶) ↔ (𝐵𝐹𝐴) = (𝐶𝐹𝐴))) |
8 | caovcang.1 | . . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑇 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → ((𝑥𝐹𝑦) = (𝑥𝐹𝑧) ↔ 𝑦 = 𝑧)) | |
9 | caovcand.2 | . . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑇) | |
10 | 8, 9, 3, 5 | caovcand 7350 | . 2 ⊢ (𝜑 → ((𝐴𝐹𝐵) = (𝐴𝐹𝐶) ↔ 𝐵 = 𝐶)) |
11 | 7, 10 | bitr3d 283 | 1 ⊢ (𝜑 → ((𝐵𝐹𝐴) = (𝐶𝐹𝐴) ↔ 𝐵 = 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 (class class class)co 7156 |
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-ral 3143 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-uni 4839 df-br 5067 df-iota 6314 df-fv 6363 df-ov 7159 |
This theorem is referenced by: (None) |
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