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Mirrors > Home > MPE Home > Th. List > 19.40b | Structured version Visualization version GIF version |
Description: The antecedent provides a condition implying the converse of 19.40 1785. This is to 19.40 1785 what 19.33b 1802 is to 19.33 1801. (Contributed by BJ, 6-May-2019.) (Proof shortened by Wolf Lammen, 13-Nov-2020.) |
Ref | Expression |
---|---|
19.40b | ⊢ ((∀𝑥𝜑 ∨ ∀𝑥𝜓) → ((∃𝑥𝜑 ∧ ∃𝑥𝜓) ↔ ∃𝑥(𝜑 ∧ 𝜓))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pm3.21 463 | . . . . 5 ⊢ (𝜓 → (𝜑 → (𝜑 ∧ 𝜓))) | |
2 | 1 | aleximi 1749 | . . . 4 ⊢ (∀𝑥𝜓 → (∃𝑥𝜑 → ∃𝑥(𝜑 ∧ 𝜓))) |
3 | pm3.2 462 | . . . . 5 ⊢ (𝜑 → (𝜓 → (𝜑 ∧ 𝜓))) | |
4 | 3 | aleximi 1749 | . . . 4 ⊢ (∀𝑥𝜑 → (∃𝑥𝜓 → ∃𝑥(𝜑 ∧ 𝜓))) |
5 | 2, 4 | jaoa 531 | . . 3 ⊢ ((∀𝑥𝜓 ∨ ∀𝑥𝜑) → ((∃𝑥𝜑 ∧ ∃𝑥𝜓) → ∃𝑥(𝜑 ∧ 𝜓))) |
6 | 5 | orcoms 403 | . 2 ⊢ ((∀𝑥𝜑 ∨ ∀𝑥𝜓) → ((∃𝑥𝜑 ∧ ∃𝑥𝜓) → ∃𝑥(𝜑 ∧ 𝜓))) |
7 | 19.40 1785 | . 2 ⊢ (∃𝑥(𝜑 ∧ 𝜓) → (∃𝑥𝜑 ∧ ∃𝑥𝜓)) | |
8 | 6, 7 | impbid1 214 | 1 ⊢ ((∀𝑥𝜑 ∨ ∀𝑥𝜓) → ((∃𝑥𝜑 ∧ ∃𝑥𝜓) ↔ ∃𝑥(𝜑 ∧ 𝜓))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∨ wo 382 ∧ wa 383 ∀wal 1473 ∃wex 1695 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-ex 1696 |
This theorem is referenced by: (None) |
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