Theorem nfd 1546

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

Hypotheses

Ref	Expression
nfd.1	⊢ Ⅎ𝑥𝜑
nfd.2	⊢ (𝜑 → (𝜓 → ∀𝑥𝜓))

Assertion

Ref	Expression
nfd	⊢ (𝜑 → Ⅎ𝑥𝜓)

Proof of Theorem nfd

Step	Hyp	Ref	Expression
1		nfd.1	. . . 4 ⊢ Ⅎ𝑥𝜑
2	1	nfri 1542	. . 3 ⊢ (𝜑 → ∀𝑥𝜑)
3		nfd.2	. . 3 ⊢ (𝜑 → (𝜓 → ∀𝑥𝜓))
4	2, 3	alrimih 1492	. 2 ⊢ (𝜑 → ∀𝑥(𝜓 → ∀𝑥𝜓))
5		df-nf 1484	. 2 ⊢ (Ⅎ𝑥𝜓 ↔ ∀𝑥(𝜓 → ∀𝑥𝜓))
6	4, 5	sylibr 134	1 ⊢ (𝜑 → Ⅎ𝑥𝜓)

Syntax hints:   → wi 4  ∀wal 1371  Ⅎwnf 1483

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1470  ax-gen 1472  ax-4 1533

This theorem depends on definitions:  df-bi 117  df-nf 1484

This theorem is referenced by:  nfdh  1547  nfrimi  1548  nfnt  1679  cbv1h  1769  nfald  1783  a16nf  1889  dvelimALT  2038  dvelimfv  2039  nfsb4t  2042  hbeud  2076