Theorem nfd 1793

Description: Deduce that 𝑥 is not free in 𝜓 in a context. (Contributed by Wolf Lammen, 16-Sep-2021.)

Hypothesis

Ref	Expression
nfd.1	⊢ (𝜑 → (∃𝑥𝜓 → ∀𝑥𝜓))

Assertion

Ref	Expression
nfd	⊢ (𝜑 → Ⅎ𝑥𝜓)

Proof of Theorem nfd

Step	Hyp	Ref	Expression
1		nfd.1	. 2 ⊢ (𝜑 → (∃𝑥𝜓 → ∀𝑥𝜓))
2		df-nf 1787	. 2 ⊢ (Ⅎ𝑥𝜓 ↔ (∃𝑥𝜓 → ∀𝑥𝜓))
3	1, 2	sylibr 233	1 ⊢ (𝜑 → Ⅎ𝑥𝜓)

Syntax hints:   → wi 4  ∀wal 1537  ∃wex 1782  Ⅎwnf 1786

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem depends on definitions:  df-bi 206  df-nf 1787

This theorem is referenced by:  nftht  1795  nfntht  1796  nfimd  1897  nf5-1  2141  axc16nf  2255  nfald  2322  nfeqf2  2377  bj-nfald  35308