Theorem nfeud 1916

Description: Deduction version of nfeu 1919. (Contributed by NM, 15-Feb-2013.) (Revised by Mario Carneiro, 7-Oct-2016.) (Proof rewritten by Jim Kingdon, 25-May-2018.)

Hypotheses

Ref	Expression
nfeud.1	⊢ Ⅎ𝑦𝜑
nfeud.2	⊢ (𝜑 → Ⅎ𝑥𝜓)

Assertion

Ref	Expression
nfeud	⊢ (𝜑 → Ⅎ𝑥∃!𝑦𝜓)

Proof of Theorem nfeud

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfv 1421	. . 3 ⊢ Ⅎ𝑧𝜓
2	1	sb8eu 1913	. 2 ⊢ (∃!𝑦𝜓 ↔ ∃!𝑧[𝑧 / 𝑦]𝜓)
3		nfv 1421	. . 3 ⊢ Ⅎ𝑧𝜑
4		nfeud.1	. . . 4 ⊢ Ⅎ𝑦𝜑
5		nfeud.2	. . . 4 ⊢ (𝜑 → Ⅎ𝑥𝜓)
6	4, 5	nfsbd 1851	. . 3 ⊢ (𝜑 → Ⅎ𝑥[𝑧 / 𝑦]𝜓)
7	3, 6	nfeudv 1915	. 2 ⊢ (𝜑 → Ⅎ𝑥∃!𝑧[𝑧 / 𝑦]𝜓)
8	2, 7	nfxfrd 1364	1 ⊢ (𝜑 → Ⅎ𝑥∃!𝑦𝜓)

Syntax hints:   → wi 4  Ⅎwnf 1349  [wsb 1645  ∃!weu 1900

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903

This theorem is referenced by:  nfmod  1917  hbeud  1922  nfreudxy  2483