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Mirrors > Home > ILE Home > Th. List > Mathboxes > elabgf1 | GIF version |
Description: One implication of elabgf 2872. (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 13774 | . 2 ⊢ (∀𝑥(𝑥 = 𝐴 → (𝜑 → 𝜓)) → (𝐴 ∈ {𝑥 ∣ 𝜑} → 𝜓)) |
4 | elabgf1.1 | . 2 ⊢ (𝑥 = 𝐴 → (𝜑 → 𝜓)) | |
5 | 3, 4 | mpg 1444 | 1 ⊢ (𝐴 ∈ {𝑥 ∣ 𝜑} → 𝜓) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1348 Ⅎwnf 1453 ∈ wcel 2141 {cab 2156 Ⅎwnfc 2299 |
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 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-ext 2152 |
This theorem depends on definitions: df-bi 116 df-tru 1351 df-nf 1454 df-sb 1756 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-v 2732 |
This theorem is referenced by: elabf1 13777 |
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