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Mirrors > Home > NFE 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 1596 | . . 3 ⊢ ((∀xψ ∧ ∃xφ) → ∃x(ψ ∧ φ)) | |
2 | 1 | ancoms 439 | . 2 ⊢ ((∃xφ ∧ ∀xψ) → ∃x(ψ ∧ φ)) |
3 | exancom 1586 | . 2 ⊢ (∃x(φ ∧ ψ) ↔ ∃x(ψ ∧ φ)) | |
4 | 2, 3 | sylibr 203 | 1 ⊢ ((∃xφ ∧ ∀xψ) → ∃x(φ ∧ ψ)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 358 ∀wal 1540 ∃wex 1541 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1546 ax-5 1557 |
This theorem depends on definitions: df-bi 177 df-an 360 df-ex 1542 |
This theorem is referenced by: 19.29r2 1598 19.29x 1599 exan 1882 equvini 1987 eu2 2229 intab 3956 imadif 5171 ncssfin 6151 |
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