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| Description: Restricted quantifier version of ceqsalv 3520. (Contributed by NM, 21-Jun-2013.) Avoid ax-9 2117, ax-12 2176, ax-ext 2707. (Revised by SN, 8-Sep-2024.) | 
| Ref | Expression | 
|---|---|
| ceqsralv.2 | ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) | 
| Ref | Expression | 
|---|---|
| ceqsralv | ⊢ (𝐴 ∈ 𝐵 → (∀𝑥 ∈ 𝐵 (𝑥 = 𝐴 → 𝜑) ↔ 𝜓)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | ceqsralv.2 | . . . 4 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) | |
| 2 | 1 | pm5.74i 271 | . . 3 ⊢ ((𝑥 = 𝐴 → 𝜑) ↔ (𝑥 = 𝐴 → 𝜓)) | 
| 3 | 2 | ralbii 3092 | . 2 ⊢ (∀𝑥 ∈ 𝐵 (𝑥 = 𝐴 → 𝜑) ↔ ∀𝑥 ∈ 𝐵 (𝑥 = 𝐴 → 𝜓)) | 
| 4 | r19.23v 3182 | . . 3 ⊢ (∀𝑥 ∈ 𝐵 (𝑥 = 𝐴 → 𝜓) ↔ (∃𝑥 ∈ 𝐵 𝑥 = 𝐴 → 𝜓)) | |
| 5 | risset 3232 | . . . 4 ⊢ (𝐴 ∈ 𝐵 ↔ ∃𝑥 ∈ 𝐵 𝑥 = 𝐴) | |
| 6 | pm5.5 361 | . . . 4 ⊢ (∃𝑥 ∈ 𝐵 𝑥 = 𝐴 → ((∃𝑥 ∈ 𝐵 𝑥 = 𝐴 → 𝜓) ↔ 𝜓)) | |
| 7 | 5, 6 | sylbi 217 | . . 3 ⊢ (𝐴 ∈ 𝐵 → ((∃𝑥 ∈ 𝐵 𝑥 = 𝐴 → 𝜓) ↔ 𝜓)) | 
| 8 | 4, 7 | bitrid 283 | . 2 ⊢ (𝐴 ∈ 𝐵 → (∀𝑥 ∈ 𝐵 (𝑥 = 𝐴 → 𝜓) ↔ 𝜓)) | 
| 9 | 3, 8 | bitrid 283 | 1 ⊢ (𝐴 ∈ 𝐵 → (∀𝑥 ∈ 𝐵 (𝑥 = 𝐴 → 𝜑) ↔ 𝜓)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 = wceq 1539 ∈ wcel 2107 ∀wral 3060 ∃wrex 3069 | 
| 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 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1779 df-clel 2815 df-ral 3061 df-rex 3070 | 
| This theorem is referenced by: eqreu 3734 sqrt2irr 16286 acsfn 17703 ovolgelb 25516 fsuppind 42605 | 
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