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Mirrors > Home > ILE Home > Th. List > riotabidv | GIF version |
Description: Formula-building deduction 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 171 | . . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐴)) | |
2 | riotabidv.1 | . . . 4 ⊢ (𝜑 → (𝜓 ↔ 𝜒)) | |
3 | 1, 2 | anbi12d 462 | . . 3 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ∧ 𝜓) ↔ (𝑥 ∈ 𝐴 ∧ 𝜒))) |
4 | 3 | iotabidv 5067 | . 2 ⊢ (𝜑 → (℩𝑥(𝑥 ∈ 𝐴 ∧ 𝜓)) = (℩𝑥(𝑥 ∈ 𝐴 ∧ 𝜒))) |
5 | df-riota 5684 | . 2 ⊢ (℩𝑥 ∈ 𝐴 𝜓) = (℩𝑥(𝑥 ∈ 𝐴 ∧ 𝜓)) | |
6 | df-riota 5684 | . 2 ⊢ (℩𝑥 ∈ 𝐴 𝜒) = (℩𝑥(𝑥 ∈ 𝐴 ∧ 𝜒)) | |
7 | 4, 5, 6 | 3eqtr4g 2172 | 1 ⊢ (𝜑 → (℩𝑥 ∈ 𝐴 𝜓) = (℩𝑥 ∈ 𝐴 𝜒)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 = wceq 1314 ∈ wcel 1463 ℩cio 5044 ℩crio 5683 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 681 ax-5 1406 ax-7 1407 ax-gen 1408 ax-ie1 1452 ax-ie2 1453 ax-8 1465 ax-10 1466 ax-11 1467 ax-i12 1468 ax-bndl 1469 ax-4 1470 ax-17 1489 ax-i9 1493 ax-ial 1497 ax-i5r 1498 ax-ext 2097 |
This theorem depends on definitions: df-bi 116 df-tru 1317 df-nf 1420 df-sb 1719 df-clab 2102 df-cleq 2108 df-clel 2111 df-nfc 2244 df-rex 2396 df-uni 3703 df-iota 5046 df-riota 5684 |
This theorem is referenced by: riotaeqbidv 5687 csbriotag 5696 infvalti 6861 caucvgsrlemfv 7533 axcaucvglemval 7632 axcaucvglemcau 7633 subval 7877 divvalap 8347 divfnzn 9315 flval 9938 cjval 10510 sqrtrval 10664 qnumval 11708 qdenval 11709 |
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