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Metamath Proof Explorer

Theorem 19.21

Description: Theorem 19.21 of Margaris p. 90. The hypothesis can be thought of as " x is not free in ph ". See 19.21v for a version requiring fewer axioms. See also 19.21h . (Contributed by NM, 14-May-1993) (Revised by Mario Carneiro, 24-Sep-2016) df-nf changed. (Revised by Wolf Lammen, 18-Sep-2021)

		Ref	Expression
	Hypothesis	19.21.1	⊢ Ⅎ 𝑥 𝜑
	Assertion	19.21	⊢ ( ∀ 𝑥 ( 𝜑 → 𝜓 ) ↔ ( 𝜑 → ∀ 𝑥 𝜓 ) )

Proof

Step	Hyp	Ref	Expression
1		19.21.1	⊢ Ⅎ 𝑥 𝜑
2		19.21t	⊢ ( Ⅎ 𝑥 𝜑 → ( ∀ 𝑥 ( 𝜑 → 𝜓 ) ↔ ( 𝜑 → ∀ 𝑥 𝜓 ) ) )
3	1 2	ax-mp	⊢ ( ∀ 𝑥 ( 𝜑 → 𝜓 ) ↔ ( 𝜑 → ∀ 𝑥 𝜓 ) )