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Mirrors > Home > ILE Home > Th. List > Mathboxes > elabf2 | GIF version |
Description: One implication of elabf 2747. (Contributed by BJ, 21-Nov-2019.) |
Ref | Expression |
---|---|
elabf2.nf | ⊢ Ⅎ𝑥𝜓 |
elabf2.s | ⊢ 𝐴 ∈ V |
elabf2.1 | ⊢ (𝑥 = 𝐴 → (𝜓 → 𝜑)) |
Ref | Expression |
---|---|
elabf2 | ⊢ (𝜓 → 𝐴 ∈ {𝑥 ∣ 𝜑}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elabf2.s | . 2 ⊢ 𝐴 ∈ V | |
2 | nfcv 2223 | . . 3 ⊢ Ⅎ𝑥𝐴 | |
3 | elabf2.nf | . . 3 ⊢ Ⅎ𝑥𝜓 | |
4 | elabf2.1 | . . 3 ⊢ (𝑥 = 𝐴 → (𝜓 → 𝜑)) | |
5 | 2, 3, 4 | elabgf2 11022 | . 2 ⊢ (𝐴 ∈ V → (𝜓 → 𝐴 ∈ {𝑥 ∣ 𝜑})) |
6 | 1, 5 | ax-mp 7 | 1 ⊢ (𝜓 → 𝐴 ∈ {𝑥 ∣ 𝜑}) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1285 Ⅎwnf 1390 ∈ wcel 1434 {cab 2069 Vcvv 2612 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 663 ax-5 1377 ax-7 1378 ax-gen 1379 ax-ie1 1423 ax-ie2 1424 ax-8 1436 ax-10 1437 ax-11 1438 ax-i12 1439 ax-bndl 1440 ax-4 1441 ax-17 1460 ax-i9 1464 ax-ial 1468 ax-i5r 1469 ax-ext 2065 |
This theorem depends on definitions: df-bi 115 df-tru 1288 df-nf 1391 df-sb 1688 df-clab 2070 df-cleq 2076 df-clel 2079 df-nfc 2212 df-v 2614 |
This theorem is referenced by: elab2a 11026 bj-bdfindis 11184 |
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