Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > caovcang | Structured version Visualization version GIF version |
Description: Convert an operation cancellation law to class notation. (Contributed by NM, 20-Aug-1995.) (Revised by Mario Carneiro, 30-Dec-2014.) |
Ref | Expression |
---|---|
caovcang.1 | ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑇 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → ((𝑥𝐹𝑦) = (𝑥𝐹𝑧) ↔ 𝑦 = 𝑧)) |
Ref | Expression |
---|---|
caovcang | ⊢ ((𝜑 ∧ (𝐴 ∈ 𝑇 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆)) → ((𝐴𝐹𝐵) = (𝐴𝐹𝐶) ↔ 𝐵 = 𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | caovcang.1 | . . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑇 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → ((𝑥𝐹𝑦) = (𝑥𝐹𝑧) ↔ 𝑦 = 𝑧)) | |
2 | 1 | ralrimivvva 3105 | . 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝑇 ∀𝑦 ∈ 𝑆 ∀𝑧 ∈ 𝑆 ((𝑥𝐹𝑦) = (𝑥𝐹𝑧) ↔ 𝑦 = 𝑧)) |
3 | oveq1 7189 | . . . . 5 ⊢ (𝑥 = 𝐴 → (𝑥𝐹𝑦) = (𝐴𝐹𝑦)) | |
4 | oveq1 7189 | . . . . 5 ⊢ (𝑥 = 𝐴 → (𝑥𝐹𝑧) = (𝐴𝐹𝑧)) | |
5 | 3, 4 | eqeq12d 2755 | . . . 4 ⊢ (𝑥 = 𝐴 → ((𝑥𝐹𝑦) = (𝑥𝐹𝑧) ↔ (𝐴𝐹𝑦) = (𝐴𝐹𝑧))) |
6 | 5 | bibi1d 347 | . . 3 ⊢ (𝑥 = 𝐴 → (((𝑥𝐹𝑦) = (𝑥𝐹𝑧) ↔ 𝑦 = 𝑧) ↔ ((𝐴𝐹𝑦) = (𝐴𝐹𝑧) ↔ 𝑦 = 𝑧))) |
7 | oveq2 7190 | . . . . 5 ⊢ (𝑦 = 𝐵 → (𝐴𝐹𝑦) = (𝐴𝐹𝐵)) | |
8 | 7 | eqeq1d 2741 | . . . 4 ⊢ (𝑦 = 𝐵 → ((𝐴𝐹𝑦) = (𝐴𝐹𝑧) ↔ (𝐴𝐹𝐵) = (𝐴𝐹𝑧))) |
9 | eqeq1 2743 | . . . 4 ⊢ (𝑦 = 𝐵 → (𝑦 = 𝑧 ↔ 𝐵 = 𝑧)) | |
10 | 8, 9 | bibi12d 349 | . . 3 ⊢ (𝑦 = 𝐵 → (((𝐴𝐹𝑦) = (𝐴𝐹𝑧) ↔ 𝑦 = 𝑧) ↔ ((𝐴𝐹𝐵) = (𝐴𝐹𝑧) ↔ 𝐵 = 𝑧))) |
11 | oveq2 7190 | . . . . 5 ⊢ (𝑧 = 𝐶 → (𝐴𝐹𝑧) = (𝐴𝐹𝐶)) | |
12 | 11 | eqeq2d 2750 | . . . 4 ⊢ (𝑧 = 𝐶 → ((𝐴𝐹𝐵) = (𝐴𝐹𝑧) ↔ (𝐴𝐹𝐵) = (𝐴𝐹𝐶))) |
13 | eqeq2 2751 | . . . 4 ⊢ (𝑧 = 𝐶 → (𝐵 = 𝑧 ↔ 𝐵 = 𝐶)) | |
14 | 12, 13 | bibi12d 349 | . . 3 ⊢ (𝑧 = 𝐶 → (((𝐴𝐹𝐵) = (𝐴𝐹𝑧) ↔ 𝐵 = 𝑧) ↔ ((𝐴𝐹𝐵) = (𝐴𝐹𝐶) ↔ 𝐵 = 𝐶))) |
15 | 6, 10, 14 | rspc3v 3542 | . 2 ⊢ ((𝐴 ∈ 𝑇 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆) → (∀𝑥 ∈ 𝑇 ∀𝑦 ∈ 𝑆 ∀𝑧 ∈ 𝑆 ((𝑥𝐹𝑦) = (𝑥𝐹𝑧) ↔ 𝑦 = 𝑧) → ((𝐴𝐹𝐵) = (𝐴𝐹𝐶) ↔ 𝐵 = 𝐶))) |
16 | 2, 15 | mpan9 510 | 1 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝑇 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆)) → ((𝐴𝐹𝐵) = (𝐴𝐹𝐶) ↔ 𝐵 = 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 ∧ w3a 1088 = wceq 1542 ∈ wcel 2114 ∀wral 3054 (class class class)co 7182 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-ext 2711 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3an 1090 df-tru 1545 df-ex 1787 df-sb 2075 df-clab 2718 df-cleq 2731 df-clel 2812 df-ral 3059 df-v 3402 df-un 3858 df-in 3860 df-ss 3870 df-sn 4527 df-pr 4529 df-op 4533 df-uni 4807 df-br 5041 df-iota 6307 df-fv 6357 df-ov 7185 |
This theorem is referenced by: caovcand 7378 caofcan 41519 |
Copyright terms: Public domain | W3C validator |