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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 2697 | . 2 ⊢ (𝐴 ∈ 𝐵 → 𝐴 ∈ V) | |
2 | elab4g.1 | . . 3 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) | |
3 | elab4g.2 | . . 3 ⊢ 𝐵 = {𝑥 ∣ 𝜑} | |
4 | 2, 3 | elab2g 2831 | . 2 ⊢ (𝐴 ∈ V → (𝐴 ∈ 𝐵 ↔ 𝜓)) |
5 | 1, 4 | biadan2 451 | 1 ⊢ (𝐴 ∈ 𝐵 ↔ (𝐴 ∈ V ∧ 𝜓)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 = wceq 1331 ∈ wcel 1480 {cab 2125 Vcvv 2686 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 |
This theorem depends on definitions: df-bi 116 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-v 2688 |
This theorem is referenced by: (None) |
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