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Mirrors > Home > MPE Home > Th. List > cnveqb | Structured version Visualization version GIF version |
Description: Equality theorem for converse. (Contributed by FL, 19-Sep-2011.) |
Ref | Expression |
---|---|
cnveqb | ⊢ ((Rel 𝐴 ∧ Rel 𝐵) → (𝐴 = 𝐵 ↔ ◡𝐴 = ◡𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cnveq 5328 | . 2 ⊢ (𝐴 = 𝐵 → ◡𝐴 = ◡𝐵) | |
2 | dfrel2 5618 | . . . 4 ⊢ (Rel 𝐴 ↔ ◡◡𝐴 = 𝐴) | |
3 | dfrel2 5618 | . . . . . . 7 ⊢ (Rel 𝐵 ↔ ◡◡𝐵 = 𝐵) | |
4 | cnveq 5328 | . . . . . . . . 9 ⊢ (◡𝐴 = ◡𝐵 → ◡◡𝐴 = ◡◡𝐵) | |
5 | eqeq2 2662 | . . . . . . . . 9 ⊢ (𝐵 = ◡◡𝐵 → (◡◡𝐴 = 𝐵 ↔ ◡◡𝐴 = ◡◡𝐵)) | |
6 | 4, 5 | syl5ibr 236 | . . . . . . . 8 ⊢ (𝐵 = ◡◡𝐵 → (◡𝐴 = ◡𝐵 → ◡◡𝐴 = 𝐵)) |
7 | 6 | eqcoms 2659 | . . . . . . 7 ⊢ (◡◡𝐵 = 𝐵 → (◡𝐴 = ◡𝐵 → ◡◡𝐴 = 𝐵)) |
8 | 3, 7 | sylbi 207 | . . . . . 6 ⊢ (Rel 𝐵 → (◡𝐴 = ◡𝐵 → ◡◡𝐴 = 𝐵)) |
9 | eqeq1 2655 | . . . . . . 7 ⊢ (𝐴 = ◡◡𝐴 → (𝐴 = 𝐵 ↔ ◡◡𝐴 = 𝐵)) | |
10 | 9 | imbi2d 329 | . . . . . 6 ⊢ (𝐴 = ◡◡𝐴 → ((◡𝐴 = ◡𝐵 → 𝐴 = 𝐵) ↔ (◡𝐴 = ◡𝐵 → ◡◡𝐴 = 𝐵))) |
11 | 8, 10 | syl5ibr 236 | . . . . 5 ⊢ (𝐴 = ◡◡𝐴 → (Rel 𝐵 → (◡𝐴 = ◡𝐵 → 𝐴 = 𝐵))) |
12 | 11 | eqcoms 2659 | . . . 4 ⊢ (◡◡𝐴 = 𝐴 → (Rel 𝐵 → (◡𝐴 = ◡𝐵 → 𝐴 = 𝐵))) |
13 | 2, 12 | sylbi 207 | . . 3 ⊢ (Rel 𝐴 → (Rel 𝐵 → (◡𝐴 = ◡𝐵 → 𝐴 = 𝐵))) |
14 | 13 | imp 444 | . 2 ⊢ ((Rel 𝐴 ∧ Rel 𝐵) → (◡𝐴 = ◡𝐵 → 𝐴 = 𝐵)) |
15 | 1, 14 | impbid2 216 | 1 ⊢ ((Rel 𝐴 ∧ Rel 𝐵) → (𝐴 = 𝐵 ↔ ◡𝐴 = ◡𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ wa 383 = wceq 1523 ◡ccnv 5142 Rel wrel 5148 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-sep 4814 ax-nul 4822 ax-pr 4936 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3an 1056 df-tru 1526 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-rab 2950 df-v 3233 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-nul 3949 df-if 4120 df-sn 4211 df-pr 4213 df-op 4217 df-br 4686 df-opab 4746 df-xp 5149 df-rel 5150 df-cnv 5151 |
This theorem is referenced by: cnveq0 5626 weisoeq2 6646 relexpaddg 13837 relexpaddss 38327 |
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