Theorem nfeud 2061

Description: Deduction version of nfeu 2064. (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 1542	. . 3 ⊢ Ⅎ𝑧𝜓
2	1	sb8eu 2058	. 2 ⊢ (∃!𝑦𝜓 ↔ ∃!𝑧[𝑧 / 𝑦]𝜓)
3		nfv 1542	. . 3 ⊢ Ⅎ𝑧𝜑
4		nfeud.1	. . . 4 ⊢ Ⅎ𝑦𝜑
5		nfeud.2	. . . 4 ⊢ (𝜑 → Ⅎ𝑥𝜓)
6	4, 5	nfsbd 1996	. . 3 ⊢ (𝜑 → Ⅎ𝑥[𝑧 / 𝑦]𝜓)
7	3, 6	nfeudv 2060	. 2 ⊢ (𝜑 → Ⅎ𝑥∃!𝑧[𝑧 / 𝑦]𝜓)
8	2, 7	nfxfrd 1489	1 ⊢ (𝜑 → Ⅎ𝑥∃!𝑦𝜓)

Syntax hints:   → wi 4  Ⅎwnf 1474  [wsb 1776  ∃!weu 2045

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-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-sb 1777  df-eu 2048

This theorem is referenced by:  nfmod  2062  hbeud  2067  nfreudxy  2671