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Metamath Proof Explorer

Theorem 19.3v

Description: Version of 19.3 with a disjoint variable condition, requiring fewer axioms. Any formula can be universally quantified using a variable which it does not contain. See also 19.9v . (Contributed by Anthony Hart, 13-Sep-2011) Remove dependency on ax-7 . (Revised by Wolf Lammen, 4-Dec-2017) (Proof shortened by Wolf Lammen, 20-Oct-2023)

		Ref	Expression
	Assertion	19.3v	⊢ ( ∀ 𝑥 𝜑 ↔ 𝜑 )

Proof

Step	Hyp	Ref	Expression
1		spvw	⊢ ( ∀ 𝑥 𝜑 → 𝜑 )
2		ax-5	⊢ ( 𝜑 → ∀ 𝑥 𝜑 )
3	1 2	impbii	⊢ ( ∀ 𝑥 𝜑 ↔ 𝜑 )