Theorem 19.21h 1797

Description: Theorem 19.21 of [Margaris] p. 90. The hypothesis can be thought of as "x is not free in φ." (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	nfi 1551	. 2 ⊢ Ⅎxφ
3	2	19.21 1796	1 ⊢ (∀x(φ → ψ) ↔ (φ → ∀xψ))

Syntax hints:   → wi 4   ↔ wb 176  ∀wal 1540

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-11 1746

This theorem depends on definitions:  df-bi 177  df-ex 1542  df-nf 1545

This theorem is referenced by:  hbim1  1810  ax12olem6  1932