Theorem 19.21h 2295

Description: Theorem 19.21 of [Margaris] p. 90. The hypothesis can be thought of as "𝑥 is not free in 𝜑". See also 19.21 2207 and 19.21v 1940. (Contributed by NM, 1-Aug-2017.) (Proof shortened by Wolf Lammen, 1-Jan-2018.)

Hypothesis

Ref	Expression
19.21h.1	⊢ (𝜑 → ∀𝑥𝜑)

Assertion

Ref	Expression
19.21h	⊢ (∀𝑥(𝜑 → 𝜓) ↔ (𝜑 → ∀𝑥𝜓))

Proof of Theorem 19.21h

Step	Hyp	Ref	Expression
1		19.21h.1	. . 3 ⊢ (𝜑 → ∀𝑥𝜑)
2	1	nf5i 2150	. 2 ⊢ Ⅎ𝑥𝜑
3	2	19.21 2207	1 ⊢ (∀𝑥(𝜑 → 𝜓) ↔ (𝜑 → ∀𝑥𝜓))

Syntax hints:   → wi 4   ↔ wb 208  ∀wal 1535

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-10 2145  ax-12 2177

This theorem depends on definitions:  df-bi 209  df-ex 1781  df-nf 1785

This theorem is referenced by:  hbim1  2305