Theorem nf5r 2123

Description: Consequence of the definition of not-free. (Contributed by Mario Carneiro, 26-Sep-2016.) df-nf 1748 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 2110	. 2 ⊢ (𝜑 → ∃𝑥𝜑)
2		df-nf 1748	. . 3 ⊢ (Ⅎ𝑥𝜑 ↔ (∃𝑥𝜑 → ∀𝑥𝜑))
3	2	biimpi 208	. 2 ⊢ (Ⅎ𝑥𝜑 → (∃𝑥𝜑 → ∀𝑥𝜑))
4	1, 3	syl5 34	1 ⊢ (Ⅎ𝑥𝜑 → (𝜑 → ∀𝑥𝜑))

Syntax hints:   → wi 4  ∀wal 1506  ∃wex 1743  Ⅎwnf 1747

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1759  ax-4 1773  ax-5 1870  ax-6 1929  ax-7 1966  ax-12 2107

This theorem depends on definitions:  df-bi 199  df-ex 1744  df-nf 1748

This theorem is referenced by:  nf5riOLD  2125  nf5rd  2126  19.3t  2131  sbft  2199  sbftALT  2521  bj-alrim  33570  bj-nexdt  33574  bj-cbv3tb  33597  bj-nfs1t2  33601  bj-equsal1t  33669  stdpc5t  33674  bj-axc14  33704  wl-nfeqfb  34251