Theorem nfd 1503

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 1499	. . 3 ⊢ (𝜑 → ∀𝑥𝜑)
3		nfd.2	. . 3 ⊢ (𝜑 → (𝜓 → ∀𝑥𝜓))
4	2, 3	alrimih 1445	. 2 ⊢ (𝜑 → ∀𝑥(𝜓 → ∀𝑥𝜓))
5		df-nf 1437	. 2 ⊢ (Ⅎ𝑥𝜓 ↔ ∀𝑥(𝜓 → ∀𝑥𝜓))
6	4, 5	sylibr 133	1 ⊢ (𝜑 → Ⅎ𝑥𝜓)

Syntax hints:   → wi 4  ∀wal 1329  Ⅎwnf 1436

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1423  ax-gen 1425  ax-4 1487

This theorem depends on definitions:  df-bi 116  df-nf 1437

This theorem is referenced by:  nfdh  1504  nfrimi  1505  nfnt  1634  cbv1h  1723  nfald  1733  a16nf  1838  dvelimALT  1983  dvelimfv  1984  nfsb4t  1987  hbeud  2019