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Mirrors > Home > ILE Home > Th. List > riotabidv | GIF version |
Description: Formula-building deduction rule for restricted iota. (Contributed by NM, 15-Sep-2011.) |
Ref | Expression |
---|---|
riotabidv.1 | ⊢ (𝜑 → (𝜓 ↔ 𝜒)) |
Ref | Expression |
---|---|
riotabidv | ⊢ (𝜑 → (℩𝑥 ∈ 𝐴 𝜓) = (℩𝑥 ∈ 𝐴 𝜒)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | biidd 170 | . . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐴)) | |
2 | riotabidv.1 | . . . 4 ⊢ (𝜑 → (𝜓 ↔ 𝜒)) | |
3 | 1, 2 | anbi12d 457 | . . 3 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ∧ 𝜓) ↔ (𝑥 ∈ 𝐴 ∧ 𝜒))) |
4 | 3 | iotabidv 4954 | . 2 ⊢ (𝜑 → (℩𝑥(𝑥 ∈ 𝐴 ∧ 𝜓)) = (℩𝑥(𝑥 ∈ 𝐴 ∧ 𝜒))) |
5 | df-riota 5546 | . 2 ⊢ (℩𝑥 ∈ 𝐴 𝜓) = (℩𝑥(𝑥 ∈ 𝐴 ∧ 𝜓)) | |
6 | df-riota 5546 | . 2 ⊢ (℩𝑥 ∈ 𝐴 𝜒) = (℩𝑥(𝑥 ∈ 𝐴 ∧ 𝜒)) | |
7 | 4, 5, 6 | 3eqtr4g 2140 | 1 ⊢ (𝜑 → (℩𝑥 ∈ 𝐴 𝜓) = (℩𝑥 ∈ 𝐴 𝜒)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 102 ↔ wb 103 = wceq 1285 ∈ wcel 1434 ℩cio 4931 ℩crio 5545 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 663 ax-5 1377 ax-7 1378 ax-gen 1379 ax-ie1 1423 ax-ie2 1424 ax-8 1436 ax-10 1437 ax-11 1438 ax-i12 1439 ax-bndl 1440 ax-4 1441 ax-17 1460 ax-i9 1464 ax-ial 1468 ax-i5r 1469 ax-ext 2065 |
This theorem depends on definitions: df-bi 115 df-tru 1288 df-nf 1391 df-sb 1688 df-clab 2070 df-cleq 2076 df-clel 2079 df-nfc 2212 df-rex 2359 df-uni 3628 df-iota 4933 df-riota 5546 |
This theorem is referenced by: riotaeqbidv 5549 csbriotag 5558 infvalti 6623 caucvgsrlemfv 7238 axcaucvglemval 7334 axcaucvglemcau 7335 subval 7576 divvalap 8038 divfnzn 9000 flval 9567 cjval 10105 sqrtrval 10259 qnumval 10942 qdenval 10943 |
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