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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 1846 | . 2 ⊢ (𝜑 → ∀𝑥(𝜓 ↔ 𝜒)) |
3 | iotabi 5097 | . 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 1329 = wceq 1331 ℩cio 5086 |
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 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 |
This theorem depends on definitions: df-bi 116 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-rex 2422 df-uni 3737 df-iota 5088 |
This theorem is referenced by: csbiotag 5116 dffv3g 5417 fveq1 5420 fveq2 5421 fvres 5445 csbfv12g 5457 fvco2 5490 riotaeqdv 5731 riotabidv 5732 riotabidva 5746 ovtposg 6156 shftval 10597 sumeq1 11124 sumeq2 11128 zsumdc 11153 isumclim3 11192 isumshft 11259 prodeq1f 11321 prodeq2w 11325 prodeq2 11326 |
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