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Mirrors > Home > ILE Home > Th. List > 19.29 | GIF version |
Description: Theorem 19.29 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 13-May-2011.) |
Ref | Expression |
---|---|
19.29 | ⊢ ((∀𝑥𝜑 ∧ ∃𝑥𝜓) → ∃𝑥(𝜑 ∧ 𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pm3.2 138 | . . . 4 ⊢ (𝜑 → (𝜓 → (𝜑 ∧ 𝜓))) | |
2 | 1 | alimi 1442 | . . 3 ⊢ (∀𝑥𝜑 → ∀𝑥(𝜓 → (𝜑 ∧ 𝜓))) |
3 | exim 1586 | . . 3 ⊢ (∀𝑥(𝜓 → (𝜑 ∧ 𝜓)) → (∃𝑥𝜓 → ∃𝑥(𝜑 ∧ 𝜓))) | |
4 | 2, 3 | syl 14 | . 2 ⊢ (∀𝑥𝜑 → (∃𝑥𝜓 → ∃𝑥(𝜑 ∧ 𝜓))) |
5 | 4 | imp 123 | 1 ⊢ ((∀𝑥𝜑 ∧ ∃𝑥𝜓) → ∃𝑥(𝜑 ∧ 𝜓)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ∀wal 1340 ∃wex 1479 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-5 1434 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-4 1497 ax-ial 1521 |
This theorem depends on definitions: df-bi 116 |
This theorem is referenced by: 19.29r 1608 19.29x 1610 19.35-1 1611 equs4 1712 equvini 1745 rexxfrd 4435 funimaexglem 5265 bj-inex 13624 |
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