Theorem nf5r 2189

Description: Consequence of the definition of not-free. (Contributed by Mario Carneiro, 26-Sep-2016.) df-nf 1781 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 2176	. 2 ⊢ (𝜑 → ∃𝑥𝜑)
2		id 22	. . 3 ⊢ (Ⅎ𝑥𝜑 → Ⅎ𝑥𝜑)
3	2	nfrd 1788	. 2 ⊢ (Ⅎ𝑥𝜑 → (∃𝑥𝜑 → ∀𝑥𝜑))
4	1, 3	syl5 34	1 ⊢ (Ⅎ𝑥𝜑 → (𝜑 → ∀𝑥𝜑))

Syntax hints:   → wi 4  ∀wal 1531  ∃wex 1776  Ⅎwnf 1780

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-12 2173

This theorem depends on definitions:  df-bi 209  df-ex 1777  df-nf 1781

This theorem is referenced by:  nf5rd  2192  19.3t  2197  sbft  2266  sbftALT  2589  bj-alrim  34022  bj-nexdt  34026  bj-cbv3tb  34104  bj-nfs1t2  34108  bj-equsal1t  34140  stdpc5t  34145  bj-axc14  34175  wl-nfeqfb  34770