Theorem nf3 1715

Description: An alternate definition of df-nf 1507. (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 1714	. 2 ⊢ (Ⅎ𝑥𝜑 ↔ (∃𝑥𝜑 → ∀𝑥𝜑))
2		nfe1 1542	. . . 4 ⊢ Ⅎ𝑥∃𝑥𝜑
3	2	nfri 1565	. . 3 ⊢ (∃𝑥𝜑 → ∀𝑥∃𝑥𝜑)
4	3	19.21h 1603	. 2 ⊢ (∀𝑥(∃𝑥𝜑 → 𝜑) ↔ (∃𝑥𝜑 → ∀𝑥𝜑))
5	1, 4	bitr4i 187	1 ⊢ (Ⅎ𝑥𝜑 ↔ ∀𝑥(∃𝑥𝜑 → 𝜑))

Syntax hints:   → wi 4   ↔ wb 105  ∀wal 1393  Ⅎwnf 1506  ∃wex 1538

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1493  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-4 1556  ax-ial 1580  ax-i5r 1581

This theorem depends on definitions:  df-bi 117  df-nf 1507

This theorem is referenced by:  hbe1a  2074  eusv2nf  4546