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Mirrors > Home > ILE Home > Th. List > elabg | GIF version |
Description: Membership in a class abstraction, using implicit substitution. Compare Theorem 6.13 of [Quine] p. 44. (Contributed by NM, 14-Apr-1995.) |
Ref | Expression |
---|---|
elabg.1 | ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) |
Ref | Expression |
---|---|
elabg | ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ {𝑥 ∣ 𝜑} ↔ 𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfcv 2228 | . 2 ⊢ Ⅎ𝑥𝐴 | |
2 | nfv 1466 | . 2 ⊢ Ⅎ𝑥𝜓 | |
3 | elabg.1 | . 2 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) | |
4 | 1, 2, 3 | elabgf 2758 | 1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ {𝑥 ∣ 𝜑} ↔ 𝜓)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 103 = wceq 1289 ∈ wcel 1438 {cab 2074 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 665 ax-5 1381 ax-7 1382 ax-gen 1383 ax-ie1 1427 ax-ie2 1428 ax-8 1440 ax-10 1441 ax-11 1442 ax-i12 1443 ax-bndl 1444 ax-4 1445 ax-17 1464 ax-i9 1468 ax-ial 1472 ax-i5r 1473 ax-ext 2070 |
This theorem depends on definitions: df-bi 115 df-tru 1292 df-nf 1395 df-sb 1693 df-clab 2075 df-cleq 2081 df-clel 2084 df-nfc 2217 df-v 2621 |
This theorem is referenced by: elab2g 2762 intmin3 3713 finds 4413 elxpi 4452 ovelrn 5785 indpi 6891 peano5nnnn 7417 peano5nni 8415 |
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