Theorem hbeu 2021

Description: Bound-variable hypothesis builder for uniqueness. Note that 𝑥 and 𝑦 needn't be distinct. (Contributed by NM, 8-Mar-1995.) (Proof rewritten by Jim Kingdon, 24-May-2018.)

Hypothesis

Ref	Expression
hbeu.1	⊢ (𝜑 → ∀𝑥𝜑)

Assertion

Ref	Expression
hbeu	⊢ (∃!𝑦𝜑 → ∀𝑥∃!𝑦𝜑)

Proof of Theorem hbeu

Step	Hyp	Ref	Expression
1		hbeu.1	. . . 4 ⊢ (𝜑 → ∀𝑥𝜑)
2	1	nfi 1439	. . 3 ⊢ Ⅎ𝑥𝜑
3	2	nfeu 2019	. 2 ⊢ Ⅎ𝑥∃!𝑦𝜑
4	3	nfri 1500	1 ⊢ (∃!𝑦𝜑 → ∀𝑥∃!𝑦𝜑)

Syntax hints:   → wi 4  ∀wal 1330  ∃!weu 2000

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 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516

This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-eu 2003

This theorem is referenced by:  hbmo  2039  2eu7  2094