Theorem hbeu 2040

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 1455	. . 3 ⊢ Ⅎ𝑥𝜑
3	2	nfeu 2038	. 2 ⊢ Ⅎ𝑥∃!𝑦𝜑
4	3	nfri 1512	1 ⊢ (∃!𝑦𝜑 → ∀𝑥∃!𝑦𝜑)

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:  hbmo  2058  2eu7  2113