Theorem nf6 2294

Description: An alternate definition of df-nf 1791. (Contributed by Mario Carneiro, 24-Sep-2016.)

Assertion

Ref	Expression
nf6	⊢ (Ⅎ𝑥𝜑 ↔ ∀𝑥(∃𝑥𝜑 → 𝜑))

Proof of Theorem nf6

Step	Hyp	Ref	Expression
1		df-nf 1791	. 2 ⊢ (Ⅎ𝑥𝜑 ↔ (∃𝑥𝜑 → ∀𝑥𝜑))
2		nfe1 2161	. . 3 ⊢ Ⅎ𝑥∃𝑥𝜑
3	2	19.21 2219	. 2 ⊢ (∀𝑥(∃𝑥𝜑 → 𝜑) ↔ (∃𝑥𝜑 → ∀𝑥𝜑))
4	1, 3	bitr4i 279	1 ⊢ (Ⅎ𝑥𝜑 ↔ ∀𝑥(∃𝑥𝜑 → 𝜑))

Syntax hints:   → wi 4   ↔ wb 207  ∀wal 1545  ∃wex 1786  Ⅎwnf 1790

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-10 2152  ax-12 2189

This theorem depends on definitions:  df-bi 208  df-ex 1787  df-nf 1791

This theorem is referenced by:  eusv2nf  5324  xfree  32533