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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 5880 | . 2 ⊢ (𝐴 = 𝐵 → ◡𝐴 = ◡𝐵) | |
2 | dfrel2 6198 | . . . 4 ⊢ (Rel 𝐴 ↔ ◡◡𝐴 = 𝐴) | |
3 | dfrel2 6198 | . . . . . . 7 ⊢ (Rel 𝐵 ↔ ◡◡𝐵 = 𝐵) | |
4 | cnveq 5880 | . . . . . . . . 9 ⊢ (◡𝐴 = ◡𝐵 → ◡◡𝐴 = ◡◡𝐵) | |
5 | eqeq2 2740 | . . . . . . . . 9 ⊢ (𝐵 = ◡◡𝐵 → (◡◡𝐴 = 𝐵 ↔ ◡◡𝐴 = ◡◡𝐵)) | |
6 | 4, 5 | imbitrrid 245 | . . . . . . . 8 ⊢ (𝐵 = ◡◡𝐵 → (◡𝐴 = ◡𝐵 → ◡◡𝐴 = 𝐵)) |
7 | 6 | eqcoms 2736 | . . . . . . 7 ⊢ (◡◡𝐵 = 𝐵 → (◡𝐴 = ◡𝐵 → ◡◡𝐴 = 𝐵)) |
8 | 3, 7 | sylbi 216 | . . . . . 6 ⊢ (Rel 𝐵 → (◡𝐴 = ◡𝐵 → ◡◡𝐴 = 𝐵)) |
9 | eqeq1 2732 | . . . . . . 7 ⊢ (𝐴 = ◡◡𝐴 → (𝐴 = 𝐵 ↔ ◡◡𝐴 = 𝐵)) | |
10 | 9 | imbi2d 339 | . . . . . 6 ⊢ (𝐴 = ◡◡𝐴 → ((◡𝐴 = ◡𝐵 → 𝐴 = 𝐵) ↔ (◡𝐴 = ◡𝐵 → ◡◡𝐴 = 𝐵))) |
11 | 8, 10 | imbitrrid 245 | . . . . 5 ⊢ (𝐴 = ◡◡𝐴 → (Rel 𝐵 → (◡𝐴 = ◡𝐵 → 𝐴 = 𝐵))) |
12 | 11 | eqcoms 2736 | . . . 4 ⊢ (◡◡𝐴 = 𝐴 → (Rel 𝐵 → (◡𝐴 = ◡𝐵 → 𝐴 = 𝐵))) |
13 | 2, 12 | sylbi 216 | . . 3 ⊢ (Rel 𝐴 → (Rel 𝐵 → (◡𝐴 = ◡𝐵 → 𝐴 = 𝐵))) |
14 | 13 | imp 405 | . 2 ⊢ ((Rel 𝐴 ∧ Rel 𝐵) → (◡𝐴 = ◡𝐵 → 𝐴 = 𝐵)) |
15 | 1, 14 | impbid2 225 | 1 ⊢ ((Rel 𝐴 ∧ Rel 𝐵) → (𝐴 = 𝐵 ↔ ◡𝐴 = ◡𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 = wceq 1533 ◡ccnv 5681 Rel wrel 5687 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2699 ax-sep 5303 ax-nul 5310 ax-pr 5433 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-sb 2060 df-clab 2706 df-cleq 2720 df-clel 2806 df-rab 3431 df-v 3475 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4327 df-if 4533 df-sn 4633 df-pr 4635 df-op 4639 df-br 5153 df-opab 5215 df-xp 5688 df-rel 5689 df-cnv 5690 |
This theorem is referenced by: cnveq0 6206 weisoeq2 7370 relexpaddg 15040 relexpaddss 43179 |
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