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Metamath Proof Explorer

Theorem ralbidva

Description: Formula-building rule for restricted universal quantifier (deduction form). (Contributed by NM, 4-Mar-1997) Reduce dependencies on axioms. (Revised by Wolf Lammen, 29-Dec-2019)

		Ref	Expression
	Hypothesis	ralbidva.1	⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → ( 𝜓 ↔ 𝜒 ) )
	Assertion	ralbidva	⊢ ( 𝜑 → ( ∀ 𝑥 ∈ 𝐴 𝜓 ↔ ∀ 𝑥 ∈ 𝐴 𝜒 ) )

Proof

Step	Hyp	Ref	Expression
1		ralbidva.1	⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → ( 𝜓 ↔ 𝜒 ) )
2	1	pm5.74da	⊢ ( 𝜑 → ( ( 𝑥 ∈ 𝐴 → 𝜓 ) ↔ ( 𝑥 ∈ 𝐴 → 𝜒 ) ) )
3	2	ralbidv2	⊢ ( 𝜑 → ( ∀ 𝑥 ∈ 𝐴 𝜓 ↔ ∀ 𝑥 ∈ 𝐴 𝜒 ) )