Theorem nfrd 1786

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 1779	. 2 ⊢ (Ⅎ𝑥𝜓 ↔ (∃𝑥𝜓 → ∀𝑥𝜓))
3	1, 2	sylib 217	1 ⊢ (𝜑 → (∃𝑥𝜓 → ∀𝑥𝜓))

Syntax hints:   → wi 4  ∀wal 1532  ∃wex 1774  Ⅎwnf 1778

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 1779

This theorem is referenced by:  19.38a  1835  19.38b  1836  nfimd  1890  nf5r  2183  19.9d  2192  nfald  2317  exists2  2653  eusv2i  5388  bj-nfimt  36108  bj-nfald  36610