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Mirrors > Home > ILE Home > Th. List > riotaeqdv | GIF version |
Description: Formula-building deduction for iota. (Contributed by NM, 15-Sep-2011.) |
Ref | Expression |
---|---|
riotaeqdv.1 | ⊢ (𝜑 → 𝐴 = 𝐵) |
Ref | Expression |
---|---|
riotaeqdv | ⊢ (𝜑 → (℩𝑥 ∈ 𝐴 𝜓) = (℩𝑥 ∈ 𝐵 𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | riotaeqdv.1 | . . . . 5 ⊢ (𝜑 → 𝐴 = 𝐵) | |
2 | 1 | eleq2d 2210 | . . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵)) |
3 | 2 | anbi1d 461 | . . 3 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ∧ 𝜓) ↔ (𝑥 ∈ 𝐵 ∧ 𝜓))) |
4 | 3 | iotabidv 5117 | . 2 ⊢ (𝜑 → (℩𝑥(𝑥 ∈ 𝐴 ∧ 𝜓)) = (℩𝑥(𝑥 ∈ 𝐵 ∧ 𝜓))) |
5 | df-riota 5738 | . 2 ⊢ (℩𝑥 ∈ 𝐴 𝜓) = (℩𝑥(𝑥 ∈ 𝐴 ∧ 𝜓)) | |
6 | df-riota 5738 | . 2 ⊢ (℩𝑥 ∈ 𝐵 𝜓) = (℩𝑥(𝑥 ∈ 𝐵 ∧ 𝜓)) | |
7 | 4, 5, 6 | 3eqtr4g 2198 | 1 ⊢ (𝜑 → (℩𝑥 ∈ 𝐴 𝜓) = (℩𝑥 ∈ 𝐵 𝜓)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 = wceq 1332 ∈ wcel 1481 ℩cio 5094 ℩crio 5737 |
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 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 |
This theorem depends on definitions: df-bi 116 df-tru 1335 df-nf 1438 df-sb 1737 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-rex 2423 df-uni 3745 df-iota 5096 df-riota 5738 |
This theorem is referenced by: riotaeqbidv 5741 |
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