Theorem nf6 2257

Description: An alternate definition of df-nf 1766. (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 1766	. 2 ⊢ (Ⅎ𝑥𝜑 ↔ (∃𝑥𝜑 → ∀𝑥𝜑))
2		nfe1 2120	. . 3 ⊢ Ⅎ𝑥∃𝑥𝜑
3	2	19.21 2172	. 2 ⊢ (∀𝑥(∃𝑥𝜑 → 𝜑) ↔ (∃𝑥𝜑 → ∀𝑥𝜑))
4	1, 3	bitr4i 279	1 ⊢ (Ⅎ𝑥𝜑 ↔ ∀𝑥(∃𝑥𝜑 → 𝜑))

Syntax hints:   → wi 4   ↔ wb 207  ∀wal 1520  ∃wex 1761  Ⅎwnf 1765

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-10 2112  ax-12 2141

This theorem depends on definitions:  df-bi 208  df-ex 1762  df-nf 1766

This theorem is referenced by:  eusv2nf  5187  xfree  29912