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Mirrors > Home > ILE Home > Th. List > iotabidv | GIF version |
Description: Formula-building deduction for iota. (Contributed by NM, 20-Aug-2011.) |
Ref | Expression |
---|---|
iotabidv.1 | ⊢ (𝜑 → (𝜓 ↔ 𝜒)) |
Ref | Expression |
---|---|
iotabidv | ⊢ (𝜑 → (℩𝑥𝜓) = (℩𝑥𝜒)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | iotabidv.1 | . . 3 ⊢ (𝜑 → (𝜓 ↔ 𝜒)) | |
2 | 1 | alrimiv 1854 | . 2 ⊢ (𝜑 → ∀𝑥(𝜓 ↔ 𝜒)) |
3 | iotabi 5146 | . 2 ⊢ (∀𝑥(𝜓 ↔ 𝜒) → (℩𝑥𝜓) = (℩𝑥𝜒)) | |
4 | 2, 3 | syl 14 | 1 ⊢ (𝜑 → (℩𝑥𝜓) = (℩𝑥𝜒)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 104 ∀wal 1333 = wceq 1335 ℩cio 5135 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 699 ax-5 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2139 |
This theorem depends on definitions: df-bi 116 df-tru 1338 df-nf 1441 df-sb 1743 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-rex 2441 df-uni 3775 df-iota 5137 |
This theorem is referenced by: csbiotag 5165 dffv3g 5466 fveq1 5469 fveq2 5470 fvres 5494 csbfv12g 5506 fvco2 5539 riotaeqdv 5783 riotabidv 5784 riotabidva 5798 ovtposg 6208 shftval 10736 sumeq1 11263 sumeq2 11267 zsumdc 11292 isumclim3 11331 isumshft 11398 prodeq1f 11460 prodeq2w 11464 prodeq2 11465 zproddc 11487 pcval 12186 |
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