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Mirrors > Home > ILE Home > Th. List > elab4g | GIF version |
Description: Membership in a class abstraction, using implicit substitution. (Contributed by NM, 17-Oct-2012.) |
Ref | Expression |
---|---|
elab4g.1 | ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) |
elab4g.2 | ⊢ 𝐵 = {𝑥 ∣ 𝜑} |
Ref | Expression |
---|---|
elab4g | ⊢ (𝐴 ∈ 𝐵 ↔ (𝐴 ∈ V ∧ 𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elex 2760 | . 2 ⊢ (𝐴 ∈ 𝐵 → 𝐴 ∈ V) | |
2 | elab4g.1 | . . 3 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) | |
3 | elab4g.2 | . . 3 ⊢ 𝐵 = {𝑥 ∣ 𝜑} | |
4 | 2, 3 | elab2g 2896 | . 2 ⊢ (𝐴 ∈ V → (𝐴 ∈ 𝐵 ↔ 𝜓)) |
5 | 1, 4 | biadan2 456 | 1 ⊢ (𝐴 ∈ 𝐵 ↔ (𝐴 ∈ V ∧ 𝜓)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 ↔ wb 105 = wceq 1363 ∈ wcel 2158 {cab 2173 Vcvv 2749 |
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 2169 |
This theorem depends on definitions: df-bi 117 df-tru 1366 df-nf 1471 df-sb 1773 df-clab 2174 df-cleq 2180 df-clel 2183 df-nfc 2318 df-v 2751 |
This theorem is referenced by: (None) |
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