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Mirrors > Home > ILE Home > Th. List > nfriotadxy | GIF version |
Description: Deduction version of nfriota 5420. (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 5411 | . 2 ⊢ (℩y ∈ A ψ) = (℩y(y ∈ A ∧ ψ)) | |
2 | nfriotadxy.1 | . . 3 ⊢ Ⅎyφ | |
3 | nfcv 2175 | . . . . . 6 ⊢ Ⅎxy | |
4 | 3 | a1i 9 | . . . . 5 ⊢ (φ → Ⅎxy) |
5 | nfriotadxy.3 | . . . . 5 ⊢ (φ → ℲxA) | |
6 | 4, 5 | nfeld 2190 | . . . 4 ⊢ (φ → Ⅎx y ∈ A) |
7 | nfriotadxy.2 | . . . 4 ⊢ (φ → Ⅎxψ) | |
8 | 6, 7 | nfand 1457 | . . 3 ⊢ (φ → Ⅎx(y ∈ A ∧ ψ)) |
9 | 2, 8 | nfiotadxy 4813 | . 2 ⊢ (φ → Ⅎx(℩y(y ∈ A ∧ ψ))) |
10 | 1, 9 | nfcxfrd 2173 | 1 ⊢ (φ → Ⅎx(℩y ∈ A ψ)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 Ⅎwnf 1346 ∈ wcel 1390 Ⅎwnfc 2162 ℩cio 4808 ℩crio 5410 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 629 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-10 1393 ax-11 1394 ax-i12 1395 ax-bndl 1396 ax-4 1397 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 |
This theorem depends on definitions: df-bi 110 df-tru 1245 df-nf 1347 df-sb 1643 df-clab 2024 df-cleq 2030 df-clel 2033 df-nfc 2164 df-rex 2306 df-sn 3373 df-uni 3572 df-iota 4810 df-riota 5411 |
This theorem is referenced by: nfriota 5420 |
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