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Mirrors > Home > ILE Home > Th. List > elab3gf | GIF version |
Description: Membership in a class abstraction, with a weaker antecedent than elabgf 2854. (Contributed by NM, 6-Sep-2011.) |
Ref | Expression |
---|---|
elab3gf.1 | ⊢ Ⅎ𝑥𝐴 |
elab3gf.2 | ⊢ Ⅎ𝑥𝜓 |
elab3gf.3 | ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) |
Ref | Expression |
---|---|
elab3gf | ⊢ ((𝜓 → 𝐴 ∈ 𝐵) → (𝐴 ∈ {𝑥 ∣ 𝜑} ↔ 𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elab3gf.1 | . . . 4 ⊢ Ⅎ𝑥𝐴 | |
2 | elab3gf.2 | . . . 4 ⊢ Ⅎ𝑥𝜓 | |
3 | elab3gf.3 | . . . 4 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) | |
4 | 1, 2, 3 | elabgf 2854 | . . 3 ⊢ (𝐴 ∈ {𝑥 ∣ 𝜑} → (𝐴 ∈ {𝑥 ∣ 𝜑} ↔ 𝜓)) |
5 | 4 | ibi 175 | . 2 ⊢ (𝐴 ∈ {𝑥 ∣ 𝜑} → 𝜓) |
6 | 1, 2, 3 | elabgf 2854 | . . . 4 ⊢ (𝐴 ∈ 𝐵 → (𝐴 ∈ {𝑥 ∣ 𝜑} ↔ 𝜓)) |
7 | 6 | imim2i 12 | . . 3 ⊢ ((𝜓 → 𝐴 ∈ 𝐵) → (𝜓 → (𝐴 ∈ {𝑥 ∣ 𝜑} ↔ 𝜓))) |
8 | biimpr 129 | . . 3 ⊢ ((𝐴 ∈ {𝑥 ∣ 𝜑} ↔ 𝜓) → (𝜓 → 𝐴 ∈ {𝑥 ∣ 𝜑})) | |
9 | 7, 8 | syli 37 | . 2 ⊢ ((𝜓 → 𝐴 ∈ 𝐵) → (𝜓 → 𝐴 ∈ {𝑥 ∣ 𝜑})) |
10 | 5, 9 | impbid2 142 | 1 ⊢ ((𝜓 → 𝐴 ∈ 𝐵) → (𝐴 ∈ {𝑥 ∣ 𝜑} ↔ 𝜓)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 104 = wceq 1335 Ⅎwnf 1440 ∈ wcel 2128 {cab 2143 Ⅎwnfc 2286 |
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 699 ax-5 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2139 |
This theorem depends on definitions: df-bi 116 df-tru 1338 df-nf 1441 df-sb 1743 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-v 2714 |
This theorem is referenced by: elab3g 2863 |
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