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Mirrors > Home > ILE Home > Th. List > 19.41h | GIF version |
Description: Theorem 19.41 of [Margaris] p. 90. New proofs should use 19.41 1573 instead. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.) (New usage is discouraged.) |
Ref | Expression |
---|---|
19.41h.1 | ⊢ (ψ → ∀xψ) |
Ref | Expression |
---|---|
19.41h | ⊢ (∃x(φ ∧ ψ) ↔ (∃xφ ∧ ψ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 19.40 1519 | . . 3 ⊢ (∃x(φ ∧ ψ) → (∃xφ ∧ ∃xψ)) | |
2 | 19.41h.1 | . . . . 5 ⊢ (ψ → ∀xψ) | |
3 | id 19 | . . . . 5 ⊢ (ψ → ψ) | |
4 | 2, 3 | exlimih 1481 | . . . 4 ⊢ (∃xψ → ψ) |
5 | 4 | anim2i 324 | . . 3 ⊢ ((∃xφ ∧ ∃xψ) → (∃xφ ∧ ψ)) |
6 | 1, 5 | syl 14 | . 2 ⊢ (∃x(φ ∧ ψ) → (∃xφ ∧ ψ)) |
7 | pm3.21 251 | . . . 4 ⊢ (ψ → (φ → (φ ∧ ψ))) | |
8 | 2, 7 | eximdh 1499 | . . 3 ⊢ (ψ → (∃xφ → ∃x(φ ∧ ψ))) |
9 | 8 | impcom 116 | . 2 ⊢ ((∃xφ ∧ ψ) → ∃x(φ ∧ ψ)) |
10 | 6, 9 | impbii 117 | 1 ⊢ (∃x(φ ∧ ψ) ↔ (∃xφ ∧ ψ)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 ↔ wb 98 ∀wal 1240 ∃wex 1378 |
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 1333 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-4 1397 ax-ial 1424 |
This theorem depends on definitions: df-bi 110 |
This theorem is referenced by: 19.42h 1574 sbh 1656 sbidm 1728 19.41v 1779 2exeu 1989 |
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