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| Mirrors > Home > MPE Home > Th. List > ceqsex | Structured version Visualization version GIF version | ||
| Description: Elimination of an existential quantifier, using implicit substitution. (Contributed by NM, 2-Mar-1995.) (Revised by Mario Carneiro, 10-Oct-2016.) (Proof shortened by Wolf Lammen, 22-Jan-2025.) | 
| Ref | Expression | 
|---|---|
| ceqsex.1 | ⊢ Ⅎ𝑥𝜓 | 
| ceqsex.2 | ⊢ 𝐴 ∈ V | 
| ceqsex.3 | ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) | 
| Ref | Expression | 
|---|---|
| ceqsex | ⊢ (∃𝑥(𝑥 = 𝐴 ∧ 𝜑) ↔ 𝜓) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | alinexa 1842 | . . 3 ⊢ (∀𝑥(𝑥 = 𝐴 → ¬ 𝜑) ↔ ¬ ∃𝑥(𝑥 = 𝐴 ∧ 𝜑)) | |
| 2 | ceqsex.1 | . . . . 5 ⊢ Ⅎ𝑥𝜓 | |
| 3 | 2 | nfn 1856 | . . . 4 ⊢ Ⅎ𝑥 ¬ 𝜓 | 
| 4 | ceqsex.2 | . . . 4 ⊢ 𝐴 ∈ V | |
| 5 | ceqsex.3 | . . . . 5 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) | |
| 6 | 5 | notbid 318 | . . . 4 ⊢ (𝑥 = 𝐴 → (¬ 𝜑 ↔ ¬ 𝜓)) | 
| 7 | 3, 4, 6 | ceqsal 3518 | . . 3 ⊢ (∀𝑥(𝑥 = 𝐴 → ¬ 𝜑) ↔ ¬ 𝜓) | 
| 8 | 1, 7 | bitr3i 277 | . 2 ⊢ (¬ ∃𝑥(𝑥 = 𝐴 ∧ 𝜑) ↔ ¬ 𝜓) | 
| 9 | 8 | con4bii 321 | 1 ⊢ (∃𝑥(𝑥 = 𝐴 ∧ 𝜑) ↔ 𝜓) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ¬ wn 3 → 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-or 848 df-ex 1779 df-nf 1783 df-clel 2815 | 
| This theorem is referenced by: ceqsex2 3534 | 
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