Theorem hbeud 1922

Description: Deduction version of hbeu 1921. (Contributed by NM, 15-Feb-2013.) (Proof rewritten by Jim Kingdon, 25-May-2018.)

Hypotheses

Ref	Expression
hbeud.1	⊢ (𝜑 → ∀𝑥𝜑)
hbeud.2	⊢ (𝜑 → ∀𝑦𝜑)
hbeud.3	⊢ (𝜑 → (𝜓 → ∀𝑥𝜓))

Assertion

Ref	Expression
hbeud	⊢ (𝜑 → (∃!𝑦𝜓 → ∀𝑥∃!𝑦𝜓))

Proof of Theorem hbeud

Step	Hyp	Ref	Expression
1		hbeud.2	. . . 4 ⊢ (𝜑 → ∀𝑦𝜑)
2	1	nfi 1351	. . 3 ⊢ Ⅎ𝑦𝜑
3		hbeud.1	. . . . 5 ⊢ (𝜑 → ∀𝑥𝜑)
4	3	nfi 1351	. . . 4 ⊢ Ⅎ𝑥𝜑
5		hbeud.3	. . . 4 ⊢ (𝜑 → (𝜓 → ∀𝑥𝜓))
6	4, 5	nfd 1416	. . 3 ⊢ (𝜑 → Ⅎ𝑥𝜓)
7	2, 6	nfeud 1916	. 2 ⊢ (𝜑 → Ⅎ𝑥∃!𝑦𝜓)
8	7	nfrd 1413	1 ⊢ (𝜑 → (∃!𝑦𝜓 → ∀𝑥∃!𝑦𝜓))

Syntax hints:   → wi 4  ∀wal 1241  ∃!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: (None)