Theorem nfeud 1957

Description: Deduction version of nfeu 1960. (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 1461	. . 3 ⊢ Ⅎ𝑧𝜓
2	1	sb8eu 1954	. 2 ⊢ (∃!𝑦𝜓 ↔ ∃!𝑧[𝑧 / 𝑦]𝜓)
3		nfv 1461	. . 3 ⊢ Ⅎ𝑧𝜑
4		nfeud.1	. . . 4 ⊢ Ⅎ𝑦𝜑
5		nfeud.2	. . . 4 ⊢ (𝜑 → Ⅎ𝑥𝜓)
6	4, 5	nfsbd 1892	. . 3 ⊢ (𝜑 → Ⅎ𝑥[𝑧 / 𝑦]𝜓)
7	3, 6	nfeudv 1956	. 2 ⊢ (𝜑 → Ⅎ𝑥∃!𝑧[𝑧 / 𝑦]𝜓)
8	2, 7	nfxfrd 1404	1 ⊢ (𝜑 → Ⅎ𝑥∃!𝑦𝜓)

Syntax hints:   → wi 4  Ⅎwnf 1389  [wsb 1685  ∃!weu 1941

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-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686  df-eu 1944

This theorem is referenced by:  nfmod  1958  hbeud  1963  nfreudxy  2527