Theorem r19.26

Description: Restricted quantifier version of 19.26 . (Contributed by NM, 28-Jan-1997) (Proof shortened by Andrew Salmon, 30-May-2011)

	Ref	Expression
Assertion	r19.26	⊢ ( ∀ 𝑥 ∈ 𝐴 ( 𝜑 ∧ 𝜓 ) ↔ ( ∀ 𝑥 ∈ 𝐴 𝜑 ∧ ∀ 𝑥 ∈ 𝐴 𝜓 ) )

Proof

Step	Hyp	Ref	Expression
1		simpl	⊢ ( ( 𝜑 ∧ 𝜓 ) → 𝜑 )
2	1	ralimi	⊢ ( ∀ 𝑥 ∈ 𝐴 ( 𝜑 ∧ 𝜓 ) → ∀ 𝑥 ∈ 𝐴 𝜑 )
3		simpr	⊢ ( ( 𝜑 ∧ 𝜓 ) → 𝜓 )
4	3	ralimi	⊢ ( ∀ 𝑥 ∈ 𝐴 ( 𝜑 ∧ 𝜓 ) → ∀ 𝑥 ∈ 𝐴 𝜓 )
5	2 4	jca	⊢ ( ∀ 𝑥 ∈ 𝐴 ( 𝜑 ∧ 𝜓 ) → ( ∀ 𝑥 ∈ 𝐴 𝜑 ∧ ∀ 𝑥 ∈ 𝐴 𝜓 ) )
6		pm3.2	⊢ ( 𝜑 → ( 𝜓 → ( 𝜑 ∧ 𝜓 ) ) )
7	6	ral2imi	⊢ ( ∀ 𝑥 ∈ 𝐴 𝜑 → ( ∀ 𝑥 ∈ 𝐴 𝜓 → ∀ 𝑥 ∈ 𝐴 ( 𝜑 ∧ 𝜓 ) ) )
8	7	imp	⊢ ( ( ∀ 𝑥 ∈ 𝐴 𝜑 ∧ ∀ 𝑥 ∈ 𝐴 𝜓 ) → ∀ 𝑥 ∈ 𝐴 ( 𝜑 ∧ 𝜓 ) )
9	5 8	impbii	⊢ ( ∀ 𝑥 ∈ 𝐴 ( 𝜑 ∧ 𝜓 ) ↔ ( ∀ 𝑥 ∈ 𝐴 𝜑 ∧ ∀ 𝑥 ∈ 𝐴 𝜓 ) )