Theorem nf3 1867

Description: An alternative definition of df-nf 1545. (Contributed by Mario Carneiro, 24-Sep-2016.)

Assertion

Ref	Expression
nf3	⊢ (Ⅎxφ ↔ ∀x(∃xφ → φ))

Proof of Theorem nf3

Step	Hyp	Ref	Expression
1		nf2 1866	. 2 ⊢ (Ⅎxφ ↔ (∃xφ → ∀xφ))
2		nfe1 1732	. . 3 ⊢ Ⅎx∃xφ
3	2	19.21 1796	. 2 ⊢ (∀x(∃xφ → φ) ↔ (∃xφ → ∀xφ))
4	1, 3	bitr4i 243	1 ⊢ (Ⅎxφ ↔ ∀x(∃xφ → φ))

Syntax hints:   → wi 4   ↔ wb 176  ∀wal 1540  ∃wex 1541  Ⅎwnf 1544

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-11 1746

This theorem depends on definitions:  df-bi 177  df-ex 1542  df-nf 1545

This theorem is referenced by: (None)