Theorem nf5 2258

Description: Alternate definition of df-nf 1770. (Contributed by Mario Carneiro, 11-Aug-2016.) df-nf 1770 changed. (Revised by Wolf Lammen, 11-Sep-2021.)

Assertion

Ref	Expression
nf5	⊢ (Ⅎ𝑥𝜑 ↔ ∀𝑥(𝜑 → ∀𝑥𝜑))

Proof of Theorem nf5

Step	Hyp	Ref	Expression
1		df-nf 1770	. 2 ⊢ (Ⅎ𝑥𝜑 ↔ (∃𝑥𝜑 → ∀𝑥𝜑))
2		nfa1 2123	. . 3 ⊢ Ⅎ𝑥∀𝑥𝜑
3	2	19.23 2178	. 2 ⊢ (∀𝑥(𝜑 → ∀𝑥𝜑) ↔ (∃𝑥𝜑 → ∀𝑥𝜑))
4	1, 3	bitr4i 279	1 ⊢ (Ⅎ𝑥𝜑 ↔ ∀𝑥(𝜑 → ∀𝑥𝜑))

Syntax hints:   → wi 4   ↔ wb 207  ∀wal 1523  ∃wex 1765  Ⅎwnf 1769

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1781  ax-4 1795  ax-5 1892  ax-6 1951  ax-7 1996  ax-10 2114  ax-12 2143

This theorem depends on definitions:  df-bi 208  df-or 843  df-ex 1766  df-nf 1770

This theorem is referenced by:  drnf1  2424  axie2  2764  xfree  29908  bj-nfdt0  33633  bj-nfalt  33649  bj-nfext  33650  bj-nfs1t  33662  bj-drnf1v  33687  bj-sbnf  33740  wl-sbnf1  34343  hbexg  40450