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Mirrors > Home > ILE Home > Th. List > elabf | GIF version |
Description: Membership in a class abstraction, using implicit substitution. (Contributed by NM, 1-Aug-1994.) (Revised by Mario Carneiro, 12-Oct-2016.) |
Ref | Expression |
---|---|
elabf.1 | ⊢ Ⅎ𝑥𝜓 |
elabf.2 | ⊢ 𝐴 ∈ V |
elabf.3 | ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) |
Ref | Expression |
---|---|
elabf | ⊢ (𝐴 ∈ {𝑥 ∣ 𝜑} ↔ 𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elabf.2 | . 2 ⊢ 𝐴 ∈ V | |
2 | nfcv 2331 | . . 3 ⊢ Ⅎ𝑥𝐴 | |
3 | elabf.1 | . . 3 ⊢ Ⅎ𝑥𝜓 | |
4 | elabf.3 | . . 3 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) | |
5 | 2, 3, 4 | elabgf 2893 | . 2 ⊢ (𝐴 ∈ V → (𝐴 ∈ {𝑥 ∣ 𝜑} ↔ 𝜓)) |
6 | 1, 5 | ax-mp 5 | 1 ⊢ (𝐴 ∈ {𝑥 ∣ 𝜑} ↔ 𝜓) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 105 = wceq 1363 Ⅎwnf 1470 ∈ wcel 2159 {cab 2174 Vcvv 2751 |
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 1457 ax-7 1458 ax-gen 1459 ax-ie1 1503 ax-ie2 1504 ax-8 1514 ax-10 1515 ax-11 1516 ax-i12 1517 ax-bndl 1519 ax-4 1520 ax-17 1536 ax-i9 1540 ax-ial 1544 ax-i5r 1545 ax-ext 2170 |
This theorem depends on definitions: df-bi 117 df-tru 1366 df-nf 1471 df-sb 1773 df-clab 2175 df-cleq 2181 df-clel 2184 df-nfc 2320 df-v 2753 |
This theorem is referenced by: elab 2895 indpi 7358 |
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