Theorem nf3 1604

Description: An alternate definition of df-nf 1395. (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 1603	. 2 ⊢ (Ⅎ𝑥𝜑 ↔ (∃𝑥𝜑 → ∀𝑥𝜑))
2		nfe1 1430	. . . 4 ⊢ Ⅎ𝑥∃𝑥𝜑
3	2	nfri 1457	. . 3 ⊢ (∃𝑥𝜑 → ∀𝑥∃𝑥𝜑)
4	3	19.21h 1494	. 2 ⊢ (∀𝑥(∃𝑥𝜑 → 𝜑) ↔ (∃𝑥𝜑 → ∀𝑥𝜑))
5	1, 4	bitr4i 185	1 ⊢ (Ⅎ𝑥𝜑 ↔ ∀𝑥(∃𝑥𝜑 → 𝜑))

Syntax hints:   → wi 4   ↔ wb 103  ∀wal 1287  Ⅎwnf 1394  ∃wex 1426

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1381  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-4 1445  ax-ial 1472  ax-i5r 1473

This theorem depends on definitions:  df-bi 115  df-nf 1395

This theorem is referenced by:  eusv2nf  4278