Theorem nf3 1647

Description: An alternate definition of df-nf 1437. (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 1646	. 2 ⊢ (Ⅎ𝑥𝜑 ↔ (∃𝑥𝜑 → ∀𝑥𝜑))
2		nfe1 1472	. . . 4 ⊢ Ⅎ𝑥∃𝑥𝜑
3	2	nfri 1499	. . 3 ⊢ (∃𝑥𝜑 → ∀𝑥∃𝑥𝜑)
4	3	19.21h 1536	. 2 ⊢ (∀𝑥(∃𝑥𝜑 → 𝜑) ↔ (∃𝑥𝜑 → ∀𝑥𝜑))
5	1, 4	bitr4i 186	1 ⊢ (Ⅎ𝑥𝜑 ↔ ∀𝑥(∃𝑥𝜑 → 𝜑))

Syntax hints:   → wi 4   ↔ wb 104  ∀wal 1329  Ⅎwnf 1436  ∃wex 1468

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1423  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-4 1487  ax-ial 1514  ax-i5r 1515

This theorem depends on definitions:  df-bi 116  df-nf 1437

This theorem is referenced by:  eusv2nf  4377