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Mirrors > Home > ILE Home > Th. List > elab3g | GIF version |
Description: Membership in a class abstraction, with a weaker antecedent than elabg 2762. (Contributed by NM, 29-Aug-2006.) |
Ref | Expression |
---|---|
elab3g.1 | ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) |
Ref | Expression |
---|---|
elab3g | ⊢ ((𝜓 → 𝐴 ∈ 𝐵) → (𝐴 ∈ {𝑥 ∣ 𝜑} ↔ 𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfcv 2229 | . 2 ⊢ Ⅎ𝑥𝐴 | |
2 | nfv 1467 | . 2 ⊢ Ⅎ𝑥𝜓 | |
3 | elab3g.1 | . 2 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) | |
4 | 1, 2, 3 | elab3gf 2766 | 1 ⊢ ((𝜓 → 𝐴 ∈ 𝐵) → (𝐴 ∈ {𝑥 ∣ 𝜑} ↔ 𝜓)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 104 = wceq 1290 ∈ wcel 1439 {cab 2075 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 666 ax-5 1382 ax-7 1383 ax-gen 1384 ax-ie1 1428 ax-ie2 1429 ax-8 1441 ax-10 1442 ax-11 1443 ax-i12 1444 ax-bndl 1445 ax-4 1446 ax-17 1465 ax-i9 1469 ax-ial 1473 ax-i5r 1474 ax-ext 2071 |
This theorem depends on definitions: df-bi 116 df-tru 1293 df-nf 1396 df-sb 1694 df-clab 2076 df-cleq 2082 df-clel 2085 df-nfc 2218 df-v 2622 |
This theorem is referenced by: elab3 2768 elssabg 3990 elrnmptg 4700 elreimasng 4811 fvelrnb 5365 elmapg 6432 |
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