Theorem nfeudv 2053

Description: Deduction version of nfeu 2057. Similar to nfeud 2054 but has the additional constraint that 𝑥 and 𝑦 must be distinct. (Contributed by Jim Kingdon, 25-May-2018.)

Hypotheses

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

Assertion

Ref	Expression
nfeudv	⊢ (𝜑 → Ⅎ𝑥∃!𝑦𝜓)

Distinct variable group:   𝑥,𝑦

Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑥,𝑦)

Proof of Theorem nfeudv

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfv 1539	. . 3 ⊢ Ⅎ𝑧𝜑
2		nfeudv.1	. . . 4 ⊢ Ⅎ𝑦𝜑
3		nfeudv.2	. . . . 5 ⊢ (𝜑 → Ⅎ𝑥𝜓)
4		nfv 1539	. . . . . 6 ⊢ Ⅎ𝑥 𝑦 = 𝑧
5	4	a1i 9	. . . . 5 ⊢ (𝜑 → Ⅎ𝑥 𝑦 = 𝑧)
6	3, 5	nfbid 1599	. . . 4 ⊢ (𝜑 → Ⅎ𝑥(𝜓 ↔ 𝑦 = 𝑧))
7	2, 6	nfald 1771	. . 3 ⊢ (𝜑 → Ⅎ𝑥∀𝑦(𝜓 ↔ 𝑦 = 𝑧))
8	1, 7	nfexd 1772	. 2 ⊢ (𝜑 → Ⅎ𝑥∃𝑧∀𝑦(𝜓 ↔ 𝑦 = 𝑧))
9		df-eu 2041	. . 3 ⊢ (∃!𝑦𝜓 ↔ ∃𝑧∀𝑦(𝜓 ↔ 𝑦 = 𝑧))
10	9	nfbii 1484	. 2 ⊢ (Ⅎ𝑥∃!𝑦𝜓 ↔ Ⅎ𝑥∃𝑧∀𝑦(𝜓 ↔ 𝑦 = 𝑧))
11	8, 10	sylibr 134	1 ⊢ (𝜑 → Ⅎ𝑥∃!𝑦𝜓)

Syntax hints:   → wi 4   ↔ wb 105  ∀wal 1362   = wceq 1364  Ⅎwnf 1471  ∃wex 1503  ∃!weu 2038

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-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-4 1521  ax-17 1537  ax-ial 1545  ax-i5r 1546

This theorem depends on definitions:  df-bi 117  df-nf 1472  df-eu 2041

This theorem is referenced by:  nfeud  2054