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Mirrors > Home > ILE Home > Th. List > 19.29r | GIF version |
Description: Variation of Theorem 19.29 of [Margaris] p. 90. (Contributed by NM, 18-Aug-1993.) |
Ref | Expression |
---|---|
19.29r | ⊢ ((∃xφ ∧ ∀xψ) → ∃x(φ ∧ ψ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 19.29 1508 | . 2 ⊢ ((∀xψ ∧ ∃xφ) → ∃x(ψ ∧ φ)) | |
2 | ancom 253 | . 2 ⊢ ((∃xφ ∧ ∀xψ) ↔ (∀xψ ∧ ∃xφ)) | |
3 | exancom 1496 | . 2 ⊢ (∃x(φ ∧ ψ) ↔ ∃x(ψ ∧ φ)) | |
4 | 1, 2, 3 | 3imtr4i 190 | 1 ⊢ ((∃xφ ∧ ∀xψ) → ∃x(φ ∧ ψ)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 ∀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: 19.29r2 1510 19.29x 1511 exan 1580 ax9o 1585 equvini 1638 eu2 1941 intab 3635 imadiflem 4921 bj-inex 9362 |
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