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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 1840 | . . 3 ⊢ (∀𝑥(𝑥 = 𝐴 → ¬ 𝜑) ↔ ¬ ∃𝑥(𝑥 = 𝐴 ∧ 𝜑)) | |
2 | ceqsex.1 | . . . . 5 ⊢ Ⅎ𝑥𝜓 | |
3 | 2 | nfn 1855 | . . . 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 3517 | . . 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 1535 = wceq 1537 ∃wex 1776 Ⅎwnf 1780 ∈ wcel 2106 Vcvv 3478 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-12 2175 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-ex 1777 df-nf 1781 df-clel 2814 |
This theorem is referenced by: ceqsexvOLDOLD 3532 ceqsex2 3535 |
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