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Mirrors > Home > ILE Home > Th. List > Mathboxes > elabgf1 | GIF version |
Description: One implication of elabgf 2826. (Contributed by BJ, 21-Nov-2019.) |
Ref | Expression |
---|---|
elabgf1.nf1 | ⊢ Ⅎ𝑥𝐴 |
elabgf1.nf2 | ⊢ Ⅎ𝑥𝜓 |
elabgf1.1 | ⊢ (𝑥 = 𝐴 → (𝜑 → 𝜓)) |
Ref | Expression |
---|---|
elabgf1 | ⊢ (𝐴 ∈ {𝑥 ∣ 𝜑} → 𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elabgf1.nf1 | . . 3 ⊢ Ⅎ𝑥𝐴 | |
2 | elabgf1.nf2 | . . 3 ⊢ Ⅎ𝑥𝜓 | |
3 | 1, 2 | elabgft1 12985 | . 2 ⊢ (∀𝑥(𝑥 = 𝐴 → (𝜑 → 𝜓)) → (𝐴 ∈ {𝑥 ∣ 𝜑} → 𝜓)) |
4 | elabgf1.1 | . 2 ⊢ (𝑥 = 𝐴 → (𝜑 → 𝜓)) | |
5 | 3, 4 | mpg 1427 | 1 ⊢ (𝐴 ∈ {𝑥 ∣ 𝜑} → 𝜓) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1331 Ⅎwnf 1436 ∈ wcel 1480 {cab 2125 Ⅎwnfc 2268 |
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 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 |
This theorem depends on definitions: df-bi 116 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-v 2688 |
This theorem is referenced by: elabf1 12988 |
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