Theorem nf5r 2233

Description: Consequence of the definition of not-free. (Contributed by Mario Carneiro, 26-Sep-2016.) df-nf 1817 changed. (Revised by Wolf Lammen, 11-Sep-2021.) (Proof shortened by Wolf Lammen, 23-Nov-2023.)

Assertion

Ref	Expression
nf5r	⊢ (Ⅎ𝑥𝜑 → (𝜑 → ∀𝑥𝜑))

Proof of Theorem nf5r

Step	Hyp	Ref	Expression
1		19.8a 2220	. 2 ⊢ (𝜑 → ∃𝑥𝜑)
2		id 23	. . 3 ⊢ (Ⅎ𝑥𝜑 → Ⅎ𝑥𝜑)
3	2	nfrd 1824	. 2 ⊢ (Ⅎ𝑥𝜑 → (∃𝑥𝜑 → ∀𝑥𝜑))
4	1, 3	syl5 35	1 ⊢ (Ⅎ𝑥𝜑 → (𝜑 → ∀𝑥𝜑))

This proof depends on syntax axioms:   → wi 4  ∀wal 1568  ∃wex 1812  Ⅎwnf 1816

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-12 2216

This proof depends on definitions:  df-bi 210  df-ex 1813  df-nf 1817

This theorem is used by:  nf5rd  2235  19.3t  2240  sbft  2307  bj-alrim  37351  bj-nexdt  37355  bj-cbv3tb  37455  bj-nfs1t2  37459  bj-equsal1t  37490  stdpc5t  37495  bj-axc14  37524  wl-nfeqfb  38224