Theorem hbeu 2035

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 1450	. . 3 ⊢ Ⅎ𝑥𝜑
3	2	nfeu 2033	. 2 ⊢ Ⅎ𝑥∃!𝑦𝜑
4	3	nfri 1507	1 ⊢ (∃!𝑦𝜑 → ∀𝑥∃!𝑦𝜑)

Syntax hints:   → wi 4  ∀wal 1341  ∃!weu 2014

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 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-eu 2017

This theorem is referenced by:  hbmo  2053  2eu7  2108