Theorem hbeud 2041

Description: Deduction version of hbeu 2040. (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 1455	. . 3 ⊢ Ⅎ𝑦𝜑
3		hbeud.1	. . . . 5 ⊢ (𝜑 → ∀𝑥𝜑)
4	3	nfi 1455	. . . 4 ⊢ Ⅎ𝑥𝜑
5		hbeud.3	. . . 4 ⊢ (𝜑 → (𝜓 → ∀𝑥𝜓))
6	4, 5	nfd 1516	. . 3 ⊢ (𝜑 → Ⅎ𝑥𝜓)
7	2, 6	nfeud 2035	. 2 ⊢ (𝜑 → Ⅎ𝑥∃!𝑦𝜓)
8	7	nfrd 1513	1 ⊢ (𝜑 → (∃!𝑦𝜓 → ∀𝑥∃!𝑦𝜓))

Syntax hints:   → wi 4  ∀wal 1346  ∃!weu 2019

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022

This theorem is referenced by: (None)