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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 2321 | . . . 4 ⊢ Ⅎ𝑥{𝑥 ∣ 𝜑} | |
3 | 1, 2 | nfel 2328 | . . 3 ⊢ Ⅎ𝑥 𝐴 ∈ {𝑥 ∣ 𝜑} |
4 | elabgf.2 | . . 3 ⊢ Ⅎ𝑥𝜓 | |
5 | 3, 4 | nfbi 1589 | . 2 ⊢ Ⅎ𝑥(𝐴 ∈ {𝑥 ∣ 𝜑} ↔ 𝜓) |
6 | eleq1 2240 | . . 3 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ {𝑥 ∣ 𝜑} ↔ 𝐴 ∈ {𝑥 ∣ 𝜑})) | |
7 | elabgf.3 | . . 3 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) | |
8 | 6, 7 | bibi12d 235 | . 2 ⊢ (𝑥 = 𝐴 → ((𝑥 ∈ {𝑥 ∣ 𝜑} ↔ 𝜑) ↔ (𝐴 ∈ {𝑥 ∣ 𝜑} ↔ 𝜓))) |
9 | abid 2165 | . 2 ⊢ (𝑥 ∈ {𝑥 ∣ 𝜑} ↔ 𝜑) | |
10 | 1, 5, 8, 9 | vtoclgf 2795 | 1 ⊢ (𝐴 ∈ 𝐵 → (𝐴 ∈ {𝑥 ∣ 𝜑} ↔ 𝜓)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 105 = wceq 1353 Ⅎwnf 1460 ∈ wcel 2148 {cab 2163 Ⅎwnfc 2306 |
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 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-ext 2159 |
This theorem depends on definitions: df-bi 117 df-tru 1356 df-nf 1461 df-sb 1763 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-v 2739 |
This theorem is referenced by: elabf 2880 elabg 2883 elab3gf 2887 elrabf 2891 bj-intabssel 14401 |
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