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Mirrors > Home > ILE Home > Th. List > exan | GIF version |
Description: Place a conjunct in the scope of an existential quantifier. (Contributed by NM, 18-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.) |
Ref | Expression |
---|---|
exan.1 | ⊢ (∃𝑥𝜑 ∧ 𝜓) |
Ref | Expression |
---|---|
exan | ⊢ ∃𝑥(𝜑 ∧ 𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hbe1 1384 | . . . 4 ⊢ (∃𝑥𝜑 → ∀𝑥∃𝑥𝜑) | |
2 | 1 | 19.28h 1454 | . . 3 ⊢ (∀𝑥(∃𝑥𝜑 ∧ 𝜓) ↔ (∃𝑥𝜑 ∧ ∀𝑥𝜓)) |
3 | exan.1 | . . 3 ⊢ (∃𝑥𝜑 ∧ 𝜓) | |
4 | 2, 3 | mpgbi 1341 | . 2 ⊢ (∃𝑥𝜑 ∧ ∀𝑥𝜓) |
5 | 19.29r 1512 | . 2 ⊢ ((∃𝑥𝜑 ∧ ∀𝑥𝜓) → ∃𝑥(𝜑 ∧ 𝜓)) | |
6 | 4, 5 | ax-mp 7 | 1 ⊢ ∃𝑥(𝜑 ∧ 𝜓) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 97 ∀wal 1241 ∃wex 1381 |
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-4 1400 ax-ial 1427 |
This theorem depends on definitions: df-bi 110 |
This theorem is referenced by: bm1.3ii 3878 bdbm1.3ii 10011 |
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