Theorem 19.28

Description: Theorem 19.28 of Margaris p. 90. See 19.28v for a version requiring fewer axioms. (Contributed by NM, 1-Aug-1993)

	Ref	Expression
Hypothesis	19.28.1	⊢ Ⅎ 𝑥 𝜑
Assertion	19.28	⊢ ( ∀ 𝑥 ( 𝜑 ∧ 𝜓 ) ↔ ( 𝜑 ∧ ∀ 𝑥 𝜓 ) )

Proof

Step	Hyp	Ref	Expression
1		19.28.1	⊢ Ⅎ 𝑥 𝜑
2		19.26	⊢ ( ∀ 𝑥 ( 𝜑 ∧ 𝜓 ) ↔ ( ∀ 𝑥 𝜑 ∧ ∀ 𝑥 𝜓 ) )
3	1	19.3	⊢ ( ∀ 𝑥 𝜑 ↔ 𝜑 )
4	3	anbi1i	⊢ ( ( ∀ 𝑥 𝜑 ∧ ∀ 𝑥 𝜓 ) ↔ ( 𝜑 ∧ ∀ 𝑥 𝜓 ) )
5	2 4	bitri	⊢ ( ∀ 𝑥 ( 𝜑 ∧ 𝜓 ) ↔ ( 𝜑 ∧ ∀ 𝑥 𝜓 ) )