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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.) |
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 | ceqsex.1 | . . 3 ⊢ Ⅎ𝑥𝜓 | |
2 | ceqsex.3 | . . . 4 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) | |
3 | 2 | biimpa 500 | . . 3 ⊢ ((𝑥 = 𝐴 ∧ 𝜑) → 𝜓) |
4 | 1, 3 | exlimi 2073 | . 2 ⊢ (∃𝑥(𝑥 = 𝐴 ∧ 𝜑) → 𝜓) |
5 | 2 | biimprcd 239 | . . . 4 ⊢ (𝜓 → (𝑥 = 𝐴 → 𝜑)) |
6 | 1, 5 | alrimi 2069 | . . 3 ⊢ (𝜓 → ∀𝑥(𝑥 = 𝐴 → 𝜑)) |
7 | ceqsex.2 | . . . 4 ⊢ 𝐴 ∈ V | |
8 | 7 | isseti 3182 | . . 3 ⊢ ∃𝑥 𝑥 = 𝐴 |
9 | exintr 1810 | . . 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 198 | 1 ⊢ (∃𝑥(𝑥 = 𝐴 ∧ 𝜑) ↔ 𝜓) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 ∀wal 1473 = wceq 1475 ∃wex 1695 Ⅎwnf 1699 ∈ wcel 1977 Vcvv 3173 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-12 2034 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-v 3175 |
This theorem is referenced by: ceqsexv 3215 ceqsex2 3217 |
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