Theorem nfeud 2096

Description: Deduction version of nfeu 2099. (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 1577	. . 3 ⊢ Ⅎ𝑧𝜓
2	1	sb8eu 2093	. 2 ⊢ (∃!𝑦𝜓 ↔ ∃!𝑧[𝑧 / 𝑦]𝜓)
3		nfv 1577	. . 3 ⊢ Ⅎ𝑧𝜑
4		nfeud.1	. . . 4 ⊢ Ⅎ𝑦𝜑
5		nfeud.2	. . . 4 ⊢ (𝜑 → Ⅎ𝑥𝜓)
6	4, 5	nfsbd 2031	. . 3 ⊢ (𝜑 → Ⅎ𝑥[𝑧 / 𝑦]𝜓)
7	3, 6	nfeudv 2095	. 2 ⊢ (𝜑 → Ⅎ𝑥∃!𝑧[𝑧 / 𝑦]𝜓)
8	2, 7	nfxfrd 1524	1 ⊢ (𝜑 → Ⅎ𝑥∃!𝑦𝜓)

Syntax hints:   → wi 4  Ⅎwnf 1509  [wsb 1811  ∃!weu 2080

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2083

This theorem is referenced by:  nfmod  2097  hbeud  2102  nfreudxy  2717