Theorem 19.40b 1921

Description: The antecedent provides a condition implying the converse of 19.40 1919. This is to 19.40 1919 what 19.33b 1918 is to 19.33 1917. (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		pm3.21 477	. . . . 5 ⊢ (𝜓 → (𝜑 → (𝜑 ∧ 𝜓)))
2	1	aleximi 1865	. . . 4 ⊢ (∀𝑥𝜓 → (∃𝑥𝜑 → ∃𝑥(𝜑 ∧ 𝜓)))
3		pm3.2 475	. . . . 5 ⊢ (𝜑 → (𝜓 → (𝜑 ∧ 𝜓)))
4	3	aleximi 1865	. . . 4 ⊢ (∀𝑥𝜑 → (∃𝑥𝜓 → ∃𝑥(𝜑 ∧ 𝜓)))
5	2, 4	jaoa 970	. . 3 ⊢ ((∀𝑥𝜓 ∨ ∀𝑥𝜑) → ((∃𝑥𝜑 ∧ ∃𝑥𝜓) → ∃𝑥(𝜑 ∧ 𝜓)))
6	5	orcoms 886	. 2 ⊢ ((∀𝑥𝜑 ∨ ∀𝑥𝜓) → ((∃𝑥𝜑 ∧ ∃𝑥𝜓) → ∃𝑥(𝜑 ∧ 𝜓)))
7		19.40 1919	. 2 ⊢ (∃𝑥(𝜑 ∧ 𝜓) → (∃𝑥𝜑 ∧ ∃𝑥𝜓))
8	6, 7	impbid1 228	1 ⊢ ((∀𝑥𝜑 ∨ ∀𝑥𝜓) → ((∃𝑥𝜑 ∧ ∃𝑥𝜓) ↔ ∃𝑥(𝜑 ∧ 𝜓)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∨ wo 861  ∀wal 1568  ∃wex 1812

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-ex 1813

This theorem is used by: (None)