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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 5737 | . 2 ⊢ (𝐴 = 𝐵 → ◡𝐴 = ◡𝐵) | |
2 | dfrel2 6039 | . . . 4 ⊢ (Rel 𝐴 ↔ ◡◡𝐴 = 𝐴) | |
3 | dfrel2 6039 | . . . . . . 7 ⊢ (Rel 𝐵 ↔ ◡◡𝐵 = 𝐵) | |
4 | cnveq 5737 | . . . . . . . . 9 ⊢ (◡𝐴 = ◡𝐵 → ◡◡𝐴 = ◡◡𝐵) | |
5 | eqeq2 2830 | . . . . . . . . 9 ⊢ (𝐵 = ◡◡𝐵 → (◡◡𝐴 = 𝐵 ↔ ◡◡𝐴 = ◡◡𝐵)) | |
6 | 4, 5 | syl5ibr 247 | . . . . . . . 8 ⊢ (𝐵 = ◡◡𝐵 → (◡𝐴 = ◡𝐵 → ◡◡𝐴 = 𝐵)) |
7 | 6 | eqcoms 2826 | . . . . . . 7 ⊢ (◡◡𝐵 = 𝐵 → (◡𝐴 = ◡𝐵 → ◡◡𝐴 = 𝐵)) |
8 | 3, 7 | sylbi 218 | . . . . . 6 ⊢ (Rel 𝐵 → (◡𝐴 = ◡𝐵 → ◡◡𝐴 = 𝐵)) |
9 | eqeq1 2822 | . . . . . . 7 ⊢ (𝐴 = ◡◡𝐴 → (𝐴 = 𝐵 ↔ ◡◡𝐴 = 𝐵)) | |
10 | 9 | imbi2d 342 | . . . . . 6 ⊢ (𝐴 = ◡◡𝐴 → ((◡𝐴 = ◡𝐵 → 𝐴 = 𝐵) ↔ (◡𝐴 = ◡𝐵 → ◡◡𝐴 = 𝐵))) |
11 | 8, 10 | syl5ibr 247 | . . . . 5 ⊢ (𝐴 = ◡◡𝐴 → (Rel 𝐵 → (◡𝐴 = ◡𝐵 → 𝐴 = 𝐵))) |
12 | 11 | eqcoms 2826 | . . . 4 ⊢ (◡◡𝐴 = 𝐴 → (Rel 𝐵 → (◡𝐴 = ◡𝐵 → 𝐴 = 𝐵))) |
13 | 2, 12 | sylbi 218 | . . 3 ⊢ (Rel 𝐴 → (Rel 𝐵 → (◡𝐴 = ◡𝐵 → 𝐴 = 𝐵))) |
14 | 13 | imp 407 | . 2 ⊢ ((Rel 𝐴 ∧ Rel 𝐵) → (◡𝐴 = ◡𝐵 → 𝐴 = 𝐵)) |
15 | 1, 14 | impbid2 227 | 1 ⊢ ((Rel 𝐴 ∧ Rel 𝐵) → (𝐴 = 𝐵 ↔ ◡𝐴 = ◡𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 = wceq 1528 ◡ccnv 5547 Rel wrel 5553 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pr 5320 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-rab 3144 df-v 3494 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-br 5058 df-opab 5120 df-xp 5554 df-rel 5555 df-cnv 5556 |
This theorem is referenced by: cnveq0 6047 weisoeq2 7098 relexpaddg 14400 relexpaddss 39941 |
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