![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > ILE Home > Th. List > nfriotadxy | GIF version |
Description: Deduction version of nfriota 5853. (Contributed by Jim Kingdon, 12-Jan-2019.) |
Ref | Expression |
---|---|
nfriotadxy.1 | ⊢ Ⅎ𝑦𝜑 |
nfriotadxy.2 | ⊢ (𝜑 → Ⅎ𝑥𝜓) |
nfriotadxy.3 | ⊢ (𝜑 → Ⅎ𝑥𝐴) |
Ref | Expression |
---|---|
nfriotadxy | ⊢ (𝜑 → Ⅎ𝑥(℩𝑦 ∈ 𝐴 𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-riota 5844 | . 2 ⊢ (℩𝑦 ∈ 𝐴 𝜓) = (℩𝑦(𝑦 ∈ 𝐴 ∧ 𝜓)) | |
2 | nfriotadxy.1 | . . 3 ⊢ Ⅎ𝑦𝜑 | |
3 | nfcv 2329 | . . . . . 6 ⊢ Ⅎ𝑥𝑦 | |
4 | 3 | a1i 9 | . . . . 5 ⊢ (𝜑 → Ⅎ𝑥𝑦) |
5 | nfriotadxy.3 | . . . . 5 ⊢ (𝜑 → Ⅎ𝑥𝐴) | |
6 | 4, 5 | nfeld 2345 | . . . 4 ⊢ (𝜑 → Ⅎ𝑥 𝑦 ∈ 𝐴) |
7 | nfriotadxy.2 | . . . 4 ⊢ (𝜑 → Ⅎ𝑥𝜓) | |
8 | 6, 7 | nfand 1578 | . . 3 ⊢ (𝜑 → Ⅎ𝑥(𝑦 ∈ 𝐴 ∧ 𝜓)) |
9 | 2, 8 | nfiotadw 5193 | . 2 ⊢ (𝜑 → Ⅎ𝑥(℩𝑦(𝑦 ∈ 𝐴 ∧ 𝜓))) |
10 | 1, 9 | nfcxfrd 2327 | 1 ⊢ (𝜑 → Ⅎ𝑥(℩𝑦 ∈ 𝐴 𝜓)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 Ⅎwnf 1470 ∈ wcel 2158 Ⅎwnfc 2316 ℩cio 5188 ℩crio 5843 |
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 710 ax-5 1457 ax-7 1458 ax-gen 1459 ax-ie1 1503 ax-ie2 1504 ax-8 1514 ax-10 1515 ax-11 1516 ax-i12 1517 ax-bndl 1519 ax-4 1520 ax-17 1536 ax-i9 1540 ax-ial 1544 ax-i5r 1545 ax-ext 2169 |
This theorem depends on definitions: df-bi 117 df-tru 1366 df-nf 1471 df-sb 1773 df-clab 2174 df-cleq 2180 df-clel 2183 df-nfc 2318 df-rex 2471 df-sn 3610 df-uni 3822 df-iota 5190 df-riota 5844 |
This theorem is referenced by: nfriota 5853 |
Copyright terms: Public domain | W3C validator |