Theorem 19.40b 1895

Description: The antecedent provides a condition implying the converse of 19.40 1893. This is to 19.40 1893 what 19.33b 1892 is to 19.33 1891. (Contributed by BJ, 6-May-2019.) (Proof shortened by Wolf Lammen, 13-Nov-2020.)

Assertion

Ref	Expression
19.40b	⊢ ((∀𝑥𝜑 ∨ ∀𝑥𝜓) → ((∃𝑥𝜑 ∧ ∃𝑥𝜓) ↔ ∃𝑥(𝜑 ∧ 𝜓)))

Proof of Theorem 19.40b

Step	Hyp	Ref	Expression
1		pm3.21 472	. . . . 5 ⊢ (𝜓 → (𝜑 → (𝜑 ∧ 𝜓)))
2	1	aleximi 1839	. . . 4 ⊢ (∀𝑥𝜓 → (∃𝑥𝜑 → ∃𝑥(𝜑 ∧ 𝜓)))
3		pm3.2 470	. . . . 5 ⊢ (𝜑 → (𝜓 → (𝜑 ∧ 𝜓)))
4	3	aleximi 1839	. . . 4 ⊢ (∀𝑥𝜑 → (∃𝑥𝜓 → ∃𝑥(𝜑 ∧ 𝜓)))
5	2, 4	jaoa 963	. . 3 ⊢ ((∀𝑥𝜓 ∨ ∀𝑥𝜑) → ((∃𝑥𝜑 ∧ ∃𝑥𝜓) → ∃𝑥(𝜑 ∧ 𝜓)))
6	5	orcoms 878	. 2 ⊢ ((∀𝑥𝜑 ∨ ∀𝑥𝜓) → ((∃𝑥𝜑 ∧ ∃𝑥𝜓) → ∃𝑥(𝜑 ∧ 𝜓)))
7		19.40 1893	. 2 ⊢ (∃𝑥(𝜑 ∧ 𝜓) → (∃𝑥𝜑 ∧ ∃𝑥𝜓))
8	6, 7	impbid1 226	1 ⊢ ((∀𝑥𝜑 ∨ ∀𝑥𝜓) → ((∃𝑥𝜑 ∧ ∃𝑥𝜓) ↔ ∃𝑥(𝜑 ∧ 𝜓)))

Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396   ∨ wo 853  ∀wal 1545  ∃wex 1786

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-ex 1787

This theorem is referenced by: (None)