Theorem hbmo 2058

Description: Bound-variable hypothesis builder for "at most one". (Contributed by NM, 9-Mar-1995.)

Hypothesis

Ref	Expression
hbmo.1	⊢ (𝜑 → ∀𝑥𝜑)

Assertion

Ref	Expression
hbmo	⊢ (∃*𝑦𝜑 → ∀𝑥∃*𝑦𝜑)

Proof of Theorem hbmo

Step	Hyp	Ref	Expression
1		df-mo 2023	. 2 ⊢ (∃*𝑦𝜑 ↔ (∃𝑦𝜑 → ∃!𝑦𝜑))
2		hbmo.1	. . . 4 ⊢ (𝜑 → ∀𝑥𝜑)
3	2	hbex 1629	. . 3 ⊢ (∃𝑦𝜑 → ∀𝑥∃𝑦𝜑)
4	2	hbeu 2040	. . 3 ⊢ (∃!𝑦𝜑 → ∀𝑥∃!𝑦𝜑)
5	3, 4	hbim 1538	. 2 ⊢ ((∃𝑦𝜑 → ∃!𝑦𝜑) → ∀𝑥(∃𝑦𝜑 → ∃!𝑦𝜑))
6	1, 5	hbxfrbi 1465	1 ⊢ (∃*𝑦𝜑 → ∀𝑥∃*𝑦𝜑)

Syntax hints:   → wi 4  ∀wal 1346  ∃wex 1485  ∃!weu 2019  ∃*wmo 2020

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  df-mo 2023

This theorem is referenced by:  moexexdc  2103  2moex  2105  2euex  2106  2exeu  2111