This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

Theorem 19.21h

Description: Theorem 19.21 of Margaris p. 90. The hypothesis can be thought of as " x is not free in ph ". See also 19.21 and 19.21v . (Contributed by NM, 1-Aug-2017) (Proof shortened by Wolf Lammen, 1-Jan-2018)

		Ref	Expression
	Hypothesis	19.21h.1	⊢ ( 𝜑 → ∀ 𝑥 𝜑 )
	Assertion	19.21h	⊢ ( ∀ 𝑥 ( 𝜑 → 𝜓 ) ↔ ( 𝜑 → ∀ 𝑥 𝜓 ) )

Proof

Step	Hyp	Ref	Expression
1		19.21h.1	⊢ ( 𝜑 → ∀ 𝑥 𝜑 )
2	1	nf5i	⊢ Ⅎ 𝑥 𝜑
3	2	19.21	⊢ ( ∀ 𝑥 ( 𝜑 → 𝜓 ) ↔ ( 𝜑 → ∀ 𝑥 𝜓 ) )