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Mirrors > Home > ILE Home > Th. List > 19.29x | GIF version |
Description: Variation of Theorem 19.29 of [Margaris] p. 90 with mixed quantification. (Contributed by NM, 11-Feb-2005.) |
Ref | Expression |
---|---|
19.29x | ⊢ ((∃x∀yφ ∧ ∀x∃yψ) → ∃x∃y(φ ∧ ψ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 19.29r 1509 | . 2 ⊢ ((∃x∀yφ ∧ ∀x∃yψ) → ∃x(∀yφ ∧ ∃yψ)) | |
2 | 19.29 1508 | . . 3 ⊢ ((∀yφ ∧ ∃yψ) → ∃y(φ ∧ ψ)) | |
3 | 2 | eximi 1488 | . 2 ⊢ (∃x(∀yφ ∧ ∃yψ) → ∃x∃y(φ ∧ ψ)) |
4 | 1, 3 | syl 14 | 1 ⊢ ((∃x∀yφ ∧ ∀x∃yψ) → ∃x∃y(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: (None) |
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