Theorem nfdv 1639

Description: Apply the definition of not-free in a context. (Contributed by Mario Carneiro, 11-Aug-2016.)

Hypothesis

Ref	Expression
nfdv.1	⊢ (φ → (ψ → ∀xψ))

Assertion

Ref	Expression
nfdv	⊢ (φ → Ⅎxψ)

Distinct variable group:   φ,x

Allowed substitution hint:   ψ(x)

Proof of Theorem nfdv

Step	Hyp	Ref	Expression
1		nfdv.1	. . 3 ⊢ (φ → (ψ → ∀xψ))
2	1	alrimiv 1631	. 2 ⊢ (φ → ∀x(ψ → ∀xψ))
3		df-nf 1545	. 2 ⊢ (Ⅎxψ ↔ ∀x(ψ → ∀xψ))
4	2, 3	sylibr 203	1 ⊢ (φ → Ⅎxψ)

This proof depends on syntax axioms:   → wi 4  ∀wal 1540  Ⅎwnf 1544

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616

This proof depends on definitions:  df-bi 177  df-nf 1545

This theorem is used by: (None)