Theorem nf5r 2218

Description: Consequence of the definition of not-free. (Contributed by Mario Carneiro, 26-Sep-2016.) df-nf 1858 changed. (Revised by Wolf Lammen, 11-Sep-2021.)

Assertion

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

Proof of Theorem nf5r

Step	Hyp	Ref	Expression
1		19.8a 2206	. 2 ⊢ (𝜑 → ∃𝑥𝜑)
2		df-nf 1858	. . 3 ⊢ (Ⅎ𝑥𝜑 ↔ (∃𝑥𝜑 → ∀𝑥𝜑))
3	2	biimpi 206	. 2 ⊢ (Ⅎ𝑥𝜑 → (∃𝑥𝜑 → ∀𝑥𝜑))
4	1, 3	syl5 34	1 ⊢ (Ⅎ𝑥𝜑 → (𝜑 → ∀𝑥𝜑))

Syntax hints:   → wi 4  ∀wal 1629  ∃wex 1852  Ⅎwnf 1856

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-12 2203

This theorem depends on definitions:  df-bi 197  df-ex 1853  df-nf 1858

This theorem is referenced by:  nf5ri  2219  nf5rd  2220  19.21tOLDOLD  2230  sbft  2526  bj-alrim  33020  bj-nexdt  33024  bj-cbv3tb  33048  bj-nfs1t2  33052  bj-sbftv  33099  bj-equsal1t  33144  stdpc5t  33149  bj-axc14  33173  wl-nfeqfb  33658