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Mirrors > Home > ILE Home > Th. List > ceqsex | 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 294 | . . 3 ⊢ ((𝑥 = 𝐴 ∧ 𝜑) → 𝜓) |
4 | 1, 3 | exlimi 1587 | . 2 ⊢ (∃𝑥(𝑥 = 𝐴 ∧ 𝜑) → 𝜓) |
5 | 2 | biimprcd 159 | . . . 4 ⊢ (𝜓 → (𝑥 = 𝐴 → 𝜑)) |
6 | 1, 5 | alrimi 1515 | . . 3 ⊢ (𝜓 → ∀𝑥(𝑥 = 𝐴 → 𝜑)) |
7 | ceqsex.2 | . . . 4 ⊢ 𝐴 ∈ V | |
8 | 7 | isseti 2738 | . . 3 ⊢ ∃𝑥 𝑥 = 𝐴 |
9 | exintr 1627 | . . 3 ⊢ (∀𝑥(𝑥 = 𝐴 → 𝜑) → (∃𝑥 𝑥 = 𝐴 → ∃𝑥(𝑥 = 𝐴 ∧ 𝜑))) | |
10 | 6, 8, 9 | mpisyl 1439 | . 2 ⊢ (𝜓 → ∃𝑥(𝑥 = 𝐴 ∧ 𝜑)) |
11 | 4, 10 | impbii 125 | 1 ⊢ (∃𝑥(𝑥 = 𝐴 ∧ 𝜑) ↔ 𝜓) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 ∀wal 1346 = wceq 1348 Ⅎwnf 1453 ∃wex 1485 ∈ wcel 2141 Vcvv 2730 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-5 1440 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-ext 2152 |
This theorem depends on definitions: df-bi 116 df-nf 1454 df-sb 1756 df-clab 2157 df-cleq 2163 df-clel 2166 df-v 2732 |
This theorem is referenced by: ceqsexv 2769 ceqsex2 2770 |
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