Theorem albidh

Description: Formula-building rule for universal quantifier (deduction form). (Contributed by NM, 26-May-1993)

	Ref	Expression
Hypotheses	albidh.1	⊢ ( 𝜑 → ∀ 𝑥 𝜑 )
	albidh.2	⊢ ( 𝜑 → ( 𝜓 ↔ 𝜒 ) )
Assertion	albidh	⊢ ( 𝜑 → ( ∀ 𝑥 𝜓 ↔ ∀ 𝑥 𝜒 ) )

Proof

Step	Hyp	Ref	Expression
1		albidh.1	⊢ ( 𝜑 → ∀ 𝑥 𝜑 )
2		albidh.2	⊢ ( 𝜑 → ( 𝜓 ↔ 𝜒 ) )
3	1 2	alrimih	⊢ ( 𝜑 → ∀ 𝑥 ( 𝜓 ↔ 𝜒 ) )
4		albi	⊢ ( ∀ 𝑥 ( 𝜓 ↔ 𝜒 ) → ( ∀ 𝑥 𝜓 ↔ ∀ 𝑥 𝜒 ) )
5	3 4	syl	⊢ ( 𝜑 → ( ∀ 𝑥 𝜓 ↔ ∀ 𝑥 𝜒 ) )