Theorem nf5r 2191

Description: Consequence of the definition of not-free. (Contributed by Mario Carneiro, 26-Sep-2016.) df-nf 1786 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 2178	. 2 ⊢ (𝜑 → ∃𝑥𝜑)
2		id 22	. . 3 ⊢ (Ⅎ𝑥𝜑 → Ⅎ𝑥𝜑)
3	2	nfrd 1793	. 2 ⊢ (Ⅎ𝑥𝜑 → (∃𝑥𝜑 → ∀𝑥𝜑))
4	1, 3	syl5 34	1 ⊢ (Ⅎ𝑥𝜑 → (𝜑 → ∀𝑥𝜑))

Syntax hints:   → wi 4  ∀wal 1536  ∃wex 1781  Ⅎwnf 1785

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-12 2175

This theorem depends on definitions:  df-bi 210  df-ex 1782  df-nf 1786

This theorem is referenced by:  nf5rd  2194  19.3t  2199  sbft  2267  sbftALT  2569  bj-alrim  34140  bj-nexdt  34144  bj-cbv3tb  34224  bj-nfs1t2  34228  bj-equsal1t  34260  stdpc5t  34265  bj-axc14  34295  wl-nfeqfb  34941