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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 1809 | . 2 ⊢ (𝜑 → ∀𝑥(𝜓 ↔ 𝜒)) |
3 | iotabi 5023 | . 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 1294 = wceq 1296 ℩cio 5012 |
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 668 ax-5 1388 ax-7 1389 ax-gen 1390 ax-ie1 1434 ax-ie2 1435 ax-8 1447 ax-10 1448 ax-11 1449 ax-i12 1450 ax-bndl 1451 ax-4 1452 ax-17 1471 ax-i9 1475 ax-ial 1479 ax-i5r 1480 ax-ext 2077 |
This theorem depends on definitions: df-bi 116 df-tru 1299 df-nf 1402 df-sb 1700 df-clab 2082 df-cleq 2088 df-clel 2091 df-nfc 2224 df-rex 2376 df-uni 3676 df-iota 5014 |
This theorem is referenced by: csbiotag 5042 dffv3g 5336 fveq1 5339 fveq2 5340 fvres 5364 csbfv12g 5375 fvco2 5408 riotaeqdv 5647 riotabidv 5648 riotabidva 5662 ovtposg 6062 shftval 10374 sumeq1 10898 sumeq2 10902 zsumdc 10927 isumclim3 10966 isumshft 11033 |
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