Theorem hbeu 2066

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 1476	. . 3 ⊢ Ⅎ𝑥𝜑
3	2	nfeu 2064	. 2 ⊢ Ⅎ𝑥∃!𝑦𝜑
4	3	nfri 1533	1 ⊢ (∃!𝑦𝜑 → ∀𝑥∃!𝑦𝜑)

Syntax hints:   → wi 4  ∀wal 1362  ∃!weu 2045

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-sb 1777  df-eu 2048

This theorem is referenced by:  hbmo  2084  2eu7  2139