Theorem nf5r 2062

Description: Consequence of the definition of not-free. (Contributed by Mario Carneiro, 26-Sep-2016.) df-nf 1708 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 2050	. 2 ⊢ (𝜑 → ∃𝑥𝜑)
2		df-nf 1708	. . 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 1479  ∃wex 1702  Ⅎwnf 1706

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-12 2045

This theorem depends on definitions:  df-bi 197  df-ex 1703  df-nf 1708

This theorem is referenced by:  nf5ri  2063  nf5rd  2064  19.21tOLDOLD  2072  sbft  2377  bj-alrim  32658  bj-nexdt  32662  bj-cbv3tb  32686  bj-nfs1t2  32690  bj-sbftv  32738  bj-equsal1t  32784  stdpc5t  32789  bj-axc14  32814  wl-nfeqfb  33294