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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 280 | . . 3 ⊢ ((𝑥 = 𝐴 ∧ 𝜑) → 𝜓) |
4 | 1, 3 | exlimi 1485 | . 2 ⊢ (∃𝑥(𝑥 = 𝐴 ∧ 𝜑) → 𝜓) |
5 | 2 | biimprcd 149 | . . . 4 ⊢ (𝜓 → (𝑥 = 𝐴 → 𝜑)) |
6 | 1, 5 | alrimi 1415 | . . 3 ⊢ (𝜓 → ∀𝑥(𝑥 = 𝐴 → 𝜑)) |
7 | ceqsex.2 | . . . 4 ⊢ 𝐴 ∈ V | |
8 | 7 | isseti 2563 | . . 3 ⊢ ∃𝑥 𝑥 = 𝐴 |
9 | exintr 1525 | . . 3 ⊢ (∀𝑥(𝑥 = 𝐴 → 𝜑) → (∃𝑥 𝑥 = 𝐴 → ∃𝑥(𝑥 = 𝐴 ∧ 𝜑))) | |
10 | 6, 8, 9 | mpisyl 1335 | . 2 ⊢ (𝜓 → ∃𝑥(𝑥 = 𝐴 ∧ 𝜑)) |
11 | 4, 10 | impbii 117 | 1 ⊢ (∃𝑥(𝑥 = 𝐴 ∧ 𝜑) ↔ 𝜓) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 ↔ wb 98 ∀wal 1241 = wceq 1243 Ⅎwnf 1349 ∃wex 1381 ∈ wcel 1393 Vcvv 2557 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-5 1336 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-4 1400 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-ext 2022 |
This theorem depends on definitions: df-bi 110 df-nf 1350 df-sb 1646 df-clab 2027 df-cleq 2033 df-clel 2036 df-v 2559 |
This theorem is referenced by: ceqsexv 2593 ceqsex2 2594 |
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