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Mirrors > Home > ILE Home > Th. List > Mathboxes > elabf2 | GIF version |
Description: One implication of elabf 2880. (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 2319 | . . 3 ⊢ Ⅎ𝑥𝐴 | |
3 | elabf2.nf | . . 3 ⊢ Ⅎ𝑥𝜓 | |
4 | elabf2.1 | . . 3 ⊢ (𝑥 = 𝐴 → (𝜓 → 𝜑)) | |
5 | 2, 3, 4 | elabgf2 14392 | . 2 ⊢ (𝐴 ∈ V → (𝜓 → 𝐴 ∈ {𝑥 ∣ 𝜑})) |
6 | 1, 5 | ax-mp 5 | 1 ⊢ (𝜓 → 𝐴 ∈ {𝑥 ∣ 𝜑}) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1353 Ⅎwnf 1460 ∈ wcel 2148 {cab 2163 Vcvv 2737 |
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 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-ext 2159 |
This theorem depends on definitions: df-bi 117 df-tru 1356 df-nf 1461 df-sb 1763 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-v 2739 |
This theorem is referenced by: elab2a 14396 bj-bdfindis 14559 |
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