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Mirrors > Home > ILE Home > Th. List > elabgf | GIF version |
Description: Membership in a class abstraction, using implicit substitution. Compare Theorem 6.13 of [Quine] p. 44. This version has bound-variable hypotheses in place of distinct variable restrictions. (Contributed by NM, 21-Sep-2003.) (Revised by Mario Carneiro, 12-Oct-2016.) |
Ref | Expression |
---|---|
elabgf.1 | ⊢ Ⅎ𝑥𝐴 |
elabgf.2 | ⊢ Ⅎ𝑥𝜓 |
elabgf.3 | ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) |
Ref | Expression |
---|---|
elabgf | ⊢ (𝐴 ∈ 𝐵 → (𝐴 ∈ {𝑥 ∣ 𝜑} ↔ 𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elabgf.1 | . 2 ⊢ Ⅎ𝑥𝐴 | |
2 | nfab1 2180 | . . . 4 ⊢ Ⅎ𝑥{𝑥 ∣ 𝜑} | |
3 | 1, 2 | nfel 2186 | . . 3 ⊢ Ⅎ𝑥 𝐴 ∈ {𝑥 ∣ 𝜑} |
4 | elabgf.2 | . . 3 ⊢ Ⅎ𝑥𝜓 | |
5 | 3, 4 | nfbi 1481 | . 2 ⊢ Ⅎ𝑥(𝐴 ∈ {𝑥 ∣ 𝜑} ↔ 𝜓) |
6 | eleq1 2100 | . . 3 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ {𝑥 ∣ 𝜑} ↔ 𝐴 ∈ {𝑥 ∣ 𝜑})) | |
7 | elabgf.3 | . . 3 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) | |
8 | 6, 7 | bibi12d 224 | . 2 ⊢ (𝑥 = 𝐴 → ((𝑥 ∈ {𝑥 ∣ 𝜑} ↔ 𝜑) ↔ (𝐴 ∈ {𝑥 ∣ 𝜑} ↔ 𝜓))) |
9 | abid 2028 | . 2 ⊢ (𝑥 ∈ {𝑥 ∣ 𝜑} ↔ 𝜑) | |
10 | 1, 5, 8, 9 | vtoclgf 2612 | 1 ⊢ (𝐴 ∈ 𝐵 → (𝐴 ∈ {𝑥 ∣ 𝜑} ↔ 𝜓)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 98 = wceq 1243 Ⅎwnf 1349 ∈ wcel 1393 {cab 2026 Ⅎwnfc 2165 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 |
This theorem depends on definitions: df-bi 110 df-tru 1246 df-nf 1350 df-sb 1646 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-v 2559 |
This theorem is referenced by: elabf 2686 elabg 2688 elab3gf 2692 elrabf 2696 bj-intabssel 9928 |
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