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| Description: Obsolete version of ceqsex 3529 as of 22-Jan-2025. (Contributed by NM, 2-Mar-1995.) (Revised by Mario Carneiro, 10-Oct-2016.) (New usage is discouraged.) (Proof modification is discouraged.) | 
| Ref | Expression | 
|---|---|
| ceqsex.1 | ⊢ Ⅎ𝑥𝜓 | 
| ceqsex.2 | ⊢ 𝐴 ∈ V | 
| ceqsex.3 | ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) | 
| Ref | Expression | 
|---|---|
| ceqsexOLD | ⊢ (∃𝑥(𝑥 = 𝐴 ∧ 𝜑) ↔ 𝜓) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | ceqsex.1 | . . 3 ⊢ Ⅎ𝑥𝜓 | |
| 2 | ceqsex.3 | . . . 4 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) | |
| 3 | 2 | biimpa 476 | . . 3 ⊢ ((𝑥 = 𝐴 ∧ 𝜑) → 𝜓) | 
| 4 | 1, 3 | exlimi 2216 | . 2 ⊢ (∃𝑥(𝑥 = 𝐴 ∧ 𝜑) → 𝜓) | 
| 5 | 2 | biimprcd 250 | . . . 4 ⊢ (𝜓 → (𝑥 = 𝐴 → 𝜑)) | 
| 6 | 1, 5 | alrimi 2212 | . . 3 ⊢ (𝜓 → ∀𝑥(𝑥 = 𝐴 → 𝜑)) | 
| 7 | ceqsex.2 | . . . 4 ⊢ 𝐴 ∈ V | |
| 8 | 7 | isseti 3497 | . . 3 ⊢ ∃𝑥 𝑥 = 𝐴 | 
| 9 | exintr 1891 | . . 3 ⊢ (∀𝑥(𝑥 = 𝐴 → 𝜑) → (∃𝑥 𝑥 = 𝐴 → ∃𝑥(𝑥 = 𝐴 ∧ 𝜑))) | |
| 10 | 6, 8, 9 | mpisyl 21 | . 2 ⊢ (𝜓 → ∃𝑥(𝑥 = 𝐴 ∧ 𝜑)) | 
| 11 | 4, 10 | impbii 209 | 1 ⊢ (∃𝑥(𝑥 = 𝐴 ∧ 𝜑) ↔ 𝜓) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∀wal 1537 = wceq 1539 ∃wex 1778 Ⅎwnf 1782 ∈ wcel 2107 Vcvv 3479 | 
| 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-12 2176 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1779 df-nf 1783 df-clel 2815 | 
| This theorem is referenced by: (None) | 
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