Theorem nfrd 1788

Description: Consequence of the definition of not-free in a context. (Contributed by Wolf Lammen, 15-Oct-2021.)

Hypothesis

Ref	Expression
nfrd.1	⊢ (𝜑 → Ⅎ𝑥𝜓)

Assertion

Ref	Expression
nfrd	⊢ (𝜑 → (∃𝑥𝜓 → ∀𝑥𝜓))

Proof of Theorem nfrd

Step	Hyp	Ref	Expression
1		nfrd.1	. 2 ⊢ (𝜑 → Ⅎ𝑥𝜓)
2		df-nf 1781	. 2 ⊢ (Ⅎ𝑥𝜓 ↔ (∃𝑥𝜓 → ∀𝑥𝜓))
3	1, 2	sylib 220	1 ⊢ (𝜑 → (∃𝑥𝜓 → ∀𝑥𝜓))

Syntax hints:   → wi 4  ∀wal 1531  ∃wex 1776  Ⅎwnf 1780

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 209  df-nf 1781

This theorem is referenced by:  19.38a  1836  19.38b  1837  nfimd  1891  nf5r  2189  19.9d  2199  nfald  2343  exists2  2745  eusv2i  5294  bj-nfimt  33971  bj-nfald  34428