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| Description: Membership in a class abstraction, with a weaker antecedent than elabg 3675. (Contributed by NM, 29-Aug-2006.) | 
| Ref | Expression | 
|---|---|
| elab3g.1 | ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) | 
| Ref | Expression | 
|---|---|
| elab3g | ⊢ ((𝜓 → 𝐴 ∈ 𝐵) → (𝐴 ∈ {𝑥 ∣ 𝜑} ↔ 𝜓)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | elab3g.1 | . . . . 5 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) | |
| 2 | 1 | elabg 3675 | . . . 4 ⊢ (𝐴 ∈ {𝑥 ∣ 𝜑} → (𝐴 ∈ {𝑥 ∣ 𝜑} ↔ 𝜓)) | 
| 3 | 2 | ibi 267 | . . 3 ⊢ (𝐴 ∈ {𝑥 ∣ 𝜑} → 𝜓) | 
| 4 | pm2.21 123 | . . 3 ⊢ (¬ 𝜓 → (𝜓 → 𝐴 ∈ {𝑥 ∣ 𝜑})) | |
| 5 | 3, 4 | impbid2 226 | . 2 ⊢ (¬ 𝜓 → (𝐴 ∈ {𝑥 ∣ 𝜑} ↔ 𝜓)) | 
| 6 | 1 | elabg 3675 | . 2 ⊢ (𝐴 ∈ 𝐵 → (𝐴 ∈ {𝑥 ∣ 𝜑} ↔ 𝜓)) | 
| 7 | 5, 6 | ja 186 | 1 ⊢ ((𝜓 → 𝐴 ∈ 𝐵) → (𝐴 ∈ {𝑥 ∣ 𝜑} ↔ 𝜓)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 = wceq 1539 ∈ wcel 2107 {cab 2713 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-ext 2707 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1542 df-ex 1779 df-sb 2064 df-clab 2714 df-cleq 2728 df-clel 2815 | 
| This theorem is referenced by: elab3 3685 elssabg 5342 elrnmptg 5971 elrelimasn 6103 elmapg 8880 isust 24213 ellimc 25909 isismty 37809 clublem 43628 | 
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