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Metamath Proof Explorer

Theorem 19.40b

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

Ref Expression
Assertion 19.40b ( ( ∀ 𝑥 𝜑 ∨ ∀ 𝑥 𝜓 ) → ( ( ∃ 𝑥 𝜑 ∧ ∃ 𝑥 𝜓 ) ↔ ∃ 𝑥 ( 𝜑 ∧ 𝜓 ) ) )

Proof

Step Hyp Ref Expression
1 pm3.21 ⊢ ( 𝜓 → ( 𝜑 → ( 𝜑 ∧ 𝜓 ) ) )
2 1 aleximi ⊢ ( ∀ 𝑥 𝜓 → ( ∃ 𝑥 𝜑 → ∃ 𝑥 ( 𝜑 ∧ 𝜓 ) ) )
3 pm3.2 ⊢ ( 𝜑 → ( 𝜓 → ( 𝜑 ∧ 𝜓 ) ) )
4 3 aleximi ⊢ ( ∀ 𝑥 𝜑 → ( ∃ 𝑥 𝜓 → ∃ 𝑥 ( 𝜑 ∧ 𝜓 ) ) )
5 2 4 jaoa ⊢ ( ( ∀ 𝑥 𝜓 ∨ ∀ 𝑥 𝜑 ) → ( ( ∃ 𝑥 𝜑 ∧ ∃ 𝑥 𝜓 ) → ∃ 𝑥 ( 𝜑 ∧ 𝜓 ) ) )
6 5 orcoms ⊢ ( ( ∀ 𝑥 𝜑 ∨ ∀ 𝑥 𝜓 ) → ( ( ∃ 𝑥 𝜑 ∧ ∃ 𝑥 𝜓 ) → ∃ 𝑥 ( 𝜑 ∧ 𝜓 ) ) )
7 19.40 ⊢ ( ∃ 𝑥 ( 𝜑 ∧ 𝜓 ) → ( ∃ 𝑥 𝜑 ∧ ∃ 𝑥 𝜓 ) )
8 6 7 impbid1 ⊢ ( ( ∀ 𝑥 𝜑 ∨ ∀ 𝑥 𝜓 ) → ( ( ∃ 𝑥 𝜑 ∧ ∃ 𝑥 𝜓 ) ↔ ∃ 𝑥 ( 𝜑 ∧ 𝜓 ) ) )