Theorem nfd 1534

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 1530	. . 3 ⊢ (𝜑 → ∀𝑥𝜑)
3		nfd.2	. . 3 ⊢ (𝜑 → (𝜓 → ∀𝑥𝜓))
4	2, 3	alrimih 1480	. 2 ⊢ (𝜑 → ∀𝑥(𝜓 → ∀𝑥𝜓))
5		df-nf 1472	. 2 ⊢ (Ⅎ𝑥𝜓 ↔ ∀𝑥(𝜓 → ∀𝑥𝜓))
6	4, 5	sylibr 134	1 ⊢ (𝜑 → Ⅎ𝑥𝜓)

Syntax hints:   → wi 4  ∀wal 1362  Ⅎwnf 1471

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 1458  ax-gen 1460  ax-4 1521

This theorem depends on definitions:  df-bi 117  df-nf 1472

This theorem is referenced by:  nfdh  1535  nfrimi  1536  nfnt  1667  cbv1h  1757  nfald  1771  a16nf  1877  dvelimALT  2026  dvelimfv  2027  nfsb4t  2030  hbeud  2064