Theorem nfd 1788

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 1782	. 2 ⊢ (Ⅎ𝑥𝜓 ↔ (∃𝑥𝜓 → ∀𝑥𝜓))
3	1, 2	sylibr 234	1 ⊢ (𝜑 → Ⅎ𝑥𝜓)

Syntax hints:   → wi 4  ∀wal 1535  ∃wex 1777  Ⅎwnf 1781

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 1782

This theorem is referenced by:  nftht  1790  nfntht  1791  nfimd  1893  nf5-1  2145  axc16nf  2264  nfald  2332  nfeqf2  2385  bj-nfald  37103