Theorem 19.40b 1885

Description: The antecedent provides a condition implying the converse of 19.40 1883. This is to 19.40 1883 what 19.33b 1882 is to 19.33 1881. (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		pm3.21 474	. . . . 5 ⊢ (𝜓 → (𝜑 → (𝜑 ∧ 𝜓)))
2	1	aleximi 1828	. . . 4 ⊢ (∀𝑥𝜓 → (∃𝑥𝜑 → ∃𝑥(𝜑 ∧ 𝜓)))
3		pm3.2 472	. . . . 5 ⊢ (𝜑 → (𝜓 → (𝜑 ∧ 𝜓)))
4	3	aleximi 1828	. . . 4 ⊢ (∀𝑥𝜑 → (∃𝑥𝜓 → ∃𝑥(𝜑 ∧ 𝜓)))
5	2, 4	jaoa 952	. . 3 ⊢ ((∀𝑥𝜓 ∨ ∀𝑥𝜑) → ((∃𝑥𝜑 ∧ ∃𝑥𝜓) → ∃𝑥(𝜑 ∧ 𝜓)))
6	5	orcoms 868	. 2 ⊢ ((∀𝑥𝜑 ∨ ∀𝑥𝜓) → ((∃𝑥𝜑 ∧ ∃𝑥𝜓) → ∃𝑥(𝜑 ∧ 𝜓)))
7		19.40 1883	. 2 ⊢ (∃𝑥(𝜑 ∧ 𝜓) → (∃𝑥𝜑 ∧ ∃𝑥𝜓))
8	6, 7	impbid1 227	1 ⊢ ((∀𝑥𝜑 ∨ ∀𝑥𝜓) → ((∃𝑥𝜑 ∧ ∃𝑥𝜓) ↔ ∃𝑥(𝜑 ∧ 𝜓)))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   ∨ wo 843  ∀wal 1531  ∃wex 1776

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-ex 1777

This theorem is referenced by: (None)