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Mirrors > Home > ILE Home > Th. List > Mathboxes > elabgf1 | GIF version |
Description: One implication of elabgf 2891. (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 14757 | . 2 ⊢ (∀𝑥(𝑥 = 𝐴 → (𝜑 → 𝜓)) → (𝐴 ∈ {𝑥 ∣ 𝜑} → 𝜓)) |
4 | elabgf1.1 | . 2 ⊢ (𝑥 = 𝐴 → (𝜑 → 𝜓)) | |
5 | 3, 4 | mpg 1461 | 1 ⊢ (𝐴 ∈ {𝑥 ∣ 𝜑} → 𝜓) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1363 Ⅎwnf 1470 ∈ wcel 2158 {cab 2173 Ⅎwnfc 2316 |
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 710 ax-5 1457 ax-7 1458 ax-gen 1459 ax-ie1 1503 ax-ie2 1504 ax-8 1514 ax-10 1515 ax-11 1516 ax-i12 1517 ax-bndl 1519 ax-4 1520 ax-17 1536 ax-i9 1540 ax-ial 1544 ax-i5r 1545 ax-ext 2169 |
This theorem depends on definitions: df-bi 117 df-tru 1366 df-nf 1471 df-sb 1773 df-clab 2174 df-cleq 2180 df-clel 2183 df-nfc 2318 df-v 2751 |
This theorem is referenced by: elabf1 14760 |
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