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| Description: Equivalent wff's yield equal restricted class abstractions (deduction form). Version of rabbidva 3443 with disjoint variable condition replaced by nonfreeness hypothesis. (Contributed by BJ, 27-Apr-2019.) Avoid ax-10 2141, ax-11 2157. (Revised by Wolf Lammen, 14-Mar-2025.) | 
| Ref | Expression | 
|---|---|
| rabbida.n | ⊢ Ⅎ𝑥𝜑 | 
| rabbida.1 | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝜓 ↔ 𝜒)) | 
| Ref | Expression | 
|---|---|
| rabbida | ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝜓} = {𝑥 ∈ 𝐴 ∣ 𝜒}) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | rabbida.n | . 2 ⊢ Ⅎ𝑥𝜑 | |
| 2 | rabbida.1 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝜓 ↔ 𝜒)) | |
| 3 | 2 | pm5.32da 579 | . 2 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ∧ 𝜓) ↔ (𝑥 ∈ 𝐴 ∧ 𝜒))) | 
| 4 | 1, 3 | rabbida4 3462 | 1 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝜓} = {𝑥 ∈ 𝐴 ∣ 𝜒}) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 Ⅎwnf 1783 ∈ wcel 2108 {crab 3436 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-9 2118 ax-12 2177 ax-ext 2708 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1780 df-nf 1784 df-sb 2065 df-clab 2715 df-cleq 2729 df-rab 3437 | 
| This theorem is referenced by: rabbid 3464 rabeqbida 3466 smfpimltmpt 46761 smfpimltxrmptf 46773 smfpimgtmpt 46796 smfpimgtxrmptf 46799 smfrec 46804 smfsupmpt 46830 smfinflem 46832 smfinfmpt 46834 | 
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