Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 3661. (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 3661 | . . . 4 ⊢ (𝐴 ∈ {𝑥 ∣ 𝜑} → (𝐴 ∈ {𝑥 ∣ 𝜑} ↔ 𝜓)) |
5 | 4 | ibi 268 | . . 3 ⊢ (𝐴 ∈ {𝑥 ∣ 𝜑} → 𝜓) |
6 | pm2.21 123 | . . 3 ⊢ (¬ 𝜓 → (𝜓 → 𝐴 ∈ {𝑥 ∣ 𝜑})) | |
7 | 5, 6 | impbid2 227 | . 2 ⊢ (¬ 𝜓 → (𝐴 ∈ {𝑥 ∣ 𝜑} ↔ 𝜓)) |
8 | 1, 2, 3 | elabgf 3661 | . 2 ⊢ (𝐴 ∈ 𝐵 → (𝐴 ∈ {𝑥 ∣ 𝜑} ↔ 𝜓)) |
9 | 7, 8 | ja 187 | 1 ⊢ ((𝜓 → 𝐴 ∈ 𝐵) → (𝐴 ∈ {𝑥 ∣ 𝜑} ↔ 𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 207 = wceq 1528 Ⅎwnf 1775 ∈ wcel 2105 {cab 2796 Ⅎwnfc 2958 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-v 3494 |
This theorem is referenced by: elab3g 3670 |
Copyright terms: Public domain | W3C validator |