Theorem nfrd 1790

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 1783	. 2 ⊢ (Ⅎ𝑥𝜓 ↔ (∃𝑥𝜓 → ∀𝑥𝜓))
3	1, 2	sylib 218	1 ⊢ (𝜑 → (∃𝑥𝜓 → ∀𝑥𝜓))

Syntax hints:   → wi 4  ∀wal 1537  ∃wex 1778  Ⅎwnf 1782

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 1783

This theorem is referenced by:  19.38a  1839  19.38b  1840  nfimd  1893  nf5r  2193  19.9d  2202  nfald  2327  exists2  2661  eusv2i  5393  bj-nfimt  36640  bj-nfald  37139  eu6w  42691