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Mirrors > Home > MPE Home > Th. List > elab3gf | Structured version Visualization version GIF version |
Description: Membership in a class abstraction, with a weaker antecedent than elabgf 3666. (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 | . . . . 5 ⊢ Ⅎ𝑥𝐴 | |
2 | elab3gf.2 | . . . . 5 ⊢ Ⅎ𝑥𝜓 | |
3 | elab3gf.3 | . . . . 5 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) | |
4 | 1, 2, 3 | elabgf 3666 | . . . 4 ⊢ (𝐴 ∈ {𝑥 ∣ 𝜑} → (𝐴 ∈ {𝑥 ∣ 𝜑} ↔ 𝜓)) |
5 | 4 | ibi 269 | . . 3 ⊢ (𝐴 ∈ {𝑥 ∣ 𝜑} → 𝜓) |
6 | pm2.21 123 | . . 3 ⊢ (¬ 𝜓 → (𝜓 → 𝐴 ∈ {𝑥 ∣ 𝜑})) | |
7 | 5, 6 | impbid2 228 | . 2 ⊢ (¬ 𝜓 → (𝐴 ∈ {𝑥 ∣ 𝜑} ↔ 𝜓)) |
8 | 1, 2, 3 | elabgf 3666 | . 2 ⊢ (𝐴 ∈ 𝐵 → (𝐴 ∈ {𝑥 ∣ 𝜑} ↔ 𝜓)) |
9 | 7, 8 | ja 188 | 1 ⊢ ((𝜓 → 𝐴 ∈ 𝐵) → (𝐴 ∈ {𝑥 ∣ 𝜑} ↔ 𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 208 = wceq 1537 Ⅎwnf 1784 ∈ wcel 2114 {cab 2801 Ⅎwnfc 2963 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-v 3498 |
This theorem is referenced by: elab3g 3675 |
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