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Metamath Proof Explorer

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 ) )