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Mirrors > Home > ILE Home > Th. List > Mathboxes > elabgf1 | GIF version |
Description: One implication of elabgf 2894. (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 14914 | . 2 ⊢ (∀𝑥(𝑥 = 𝐴 → (𝜑 → 𝜓)) → (𝐴 ∈ {𝑥 ∣ 𝜑} → 𝜓)) |
4 | elabgf1.1 | . 2 ⊢ (𝑥 = 𝐴 → (𝜑 → 𝜓)) | |
5 | 3, 4 | mpg 1462 | 1 ⊢ (𝐴 ∈ {𝑥 ∣ 𝜑} → 𝜓) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1364 Ⅎwnf 1471 ∈ wcel 2160 {cab 2175 Ⅎwnfc 2319 |
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 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-ext 2171 |
This theorem depends on definitions: df-bi 117 df-tru 1367 df-nf 1472 df-sb 1774 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-v 2754 |
This theorem is referenced by: elabf1 14917 |
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