Theorem nfdh 1767

Description: Deduce that x is not free in ψ in a context. (Contributed by Mario Carneiro, 24-Sep-2016.)

Hypotheses

Ref	Expression
nfdh.1	⊢ (φ → ∀xφ)
nfdh.2	⊢ (φ → (ψ → ∀xψ))

Assertion

Ref	Expression
nfdh	⊢ (φ → Ⅎxψ)

Proof of Theorem nfdh

Step	Hyp	Ref	Expression
1		nfdh.1	. . 3 ⊢ (φ → ∀xφ)
2	1	nfi 1551	. 2 ⊢ Ⅎxφ
3		nfdh.2	. 2 ⊢ (φ → (ψ → ∀xψ))
4	2, 3	nfd 1766	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  ax-9 1654  ax-8 1675  ax-11 1746

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

This theorem is used by:  hbimd  1815  ax11indalem  2197  ax11inda2ALT  2198