Theorem nf5r 2236

Description: Consequence of the definition of not-free. (Contributed by Mario Carneiro, 26-Sep-2016.) df-nf 1811 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 2223	. 2 ⊢ (𝜑 → ∃𝑥𝜑)
2		id 23	. . 3 ⊢ (Ⅎ𝑥𝜑 → Ⅎ𝑥𝜑)
3	2	nfrd 1818	. 2 ⊢ (Ⅎ𝑥𝜑 → (∃𝑥𝜑 → ∀𝑥𝜑))
4	1, 3	syl5 35	1 ⊢ (Ⅎ𝑥𝜑 → (𝜑 → ∀𝑥𝜑))

Syntax hints:   → wi 4  ∀wal 1565  ∃wex 1806  Ⅎwnf 1810

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-12 2219

This theorem depends on definitions:  df-bi 210  df-ex 1807  df-nf 1811

This theorem is referenced by:  nf5rd  2238  19.3t  2243  sbft  2311  bj-alrim  37203  bj-nexdt  37207  bj-cbv3tb  37307  bj-nfs1t2  37311  bj-equsal1t  37342  stdpc5t  37347  bj-axc14  37376  wl-nfeqfb  38074