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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 1552 | . 2 ⊢ ((∃𝑥∀𝑦𝜑 ∧ ∀𝑥∃𝑦𝜓) → ∃𝑥(∀𝑦𝜑 ∧ ∃𝑦𝜓)) | |
2 | 19.29 1551 | . . 3 ⊢ ((∀𝑦𝜑 ∧ ∃𝑦𝜓) → ∃𝑦(𝜑 ∧ 𝜓)) | |
3 | 2 | eximi 1531 | . 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 102 ∀wal 1282 ∃wex 1421 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-5 1376 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-4 1440 ax-ial 1467 |
This theorem depends on definitions: df-bi 115 |
This theorem is referenced by: (None) |
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