Theorem 19.28v

Description: Version of 19.28 with a disjoint variable condition, requiring fewer axioms. (Contributed by NM, 25-Mar-2004)

	Ref	Expression
Assertion	19.28v	⊢ ( ∀ 𝑥 ( 𝜑 ∧ 𝜓 ) ↔ ( 𝜑 ∧ ∀ 𝑥 𝜓 ) )

Proof

Step	Hyp	Ref	Expression
1		19.26	⊢ ( ∀ 𝑥 ( 𝜑 ∧ 𝜓 ) ↔ ( ∀ 𝑥 𝜑 ∧ ∀ 𝑥 𝜓 ) )
2		19.3v	⊢ ( ∀ 𝑥 𝜑 ↔ 𝜑 )
3	2	anbi1i	⊢ ( ( ∀ 𝑥 𝜑 ∧ ∀ 𝑥 𝜓 ) ↔ ( 𝜑 ∧ ∀ 𝑥 𝜓 ) )
4	1 3	bitri	⊢ ( ∀ 𝑥 ( 𝜑 ∧ 𝜓 ) ↔ ( 𝜑 ∧ ∀ 𝑥 𝜓 ) )