Theorem hbeu 2107

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 1515	. . 3 ⊢ Ⅎ𝑥𝜑
3	2	nfeu 2105	. 2 ⊢ Ⅎ𝑥∃!𝑦𝜑
4	3	nfri 1572	1 ⊢ (∃!𝑦𝜑 → ∀𝑥∃!𝑦𝜑)

Syntax hints:   → wi 4  ∀wal 1400  ∃!weu 2086

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-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089

This theorem is referenced by:  hbmo  2125  2eu7  2181