Theorem nfd 1790

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 1784	. 2 ⊢ (Ⅎ𝑥𝜓 ↔ (∃𝑥𝜓 → ∀𝑥𝜓))
3	1, 2	sylibr 234	1 ⊢ (𝜑 → Ⅎ𝑥𝜓)

Syntax hints:   → wi 4  ∀wal 1538  ∃wex 1779  Ⅎwnf 1783

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 207  df-nf 1784

This theorem is referenced by:  nftht  1792  nfntht  1793  nfimd  1894  nf5-1  2145  axc16nf  2263  nfald  2328  nfeqf2  2382  bj-nfald  37138