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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 1874 | . 2 ⊢ (𝜑 → ∀𝑥(𝜓 ↔ 𝜒)) |
3 | iotabi 5189 | . 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 105 ∀wal 1351 = wceq 1353 ℩cio 5178 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-ext 2159 |
This theorem depends on definitions: df-bi 117 df-tru 1356 df-nf 1461 df-sb 1763 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-rex 2461 df-uni 3812 df-iota 5180 |
This theorem is referenced by: csbiotag 5211 dffv3g 5513 fveq1 5516 fveq2 5517 fvres 5541 csbfv12g 5554 fvco2 5588 riotaeqdv 5835 riotabidv 5836 riotabidva 5850 ovtposg 6263 shftval 10837 sumeq1 11366 sumeq2 11370 zsumdc 11395 isumclim3 11434 isumshft 11501 prodeq1f 11563 prodeq2w 11567 prodeq2 11568 zproddc 11590 pcval 12299 grpidvalg 12798 grpidpropdg 12799 dfur2g 13151 oppr0g 13257 oppr1g 13258 |
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