Theorem hbmo 2116

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 2081	. 2 ⊢ (∃*𝑦𝜑 ↔ (∃𝑦𝜑 → ∃!𝑦𝜑))
2		hbmo.1	. . . 4 ⊢ (𝜑 → ∀𝑥𝜑)
3	2	hbex 1682	. . 3 ⊢ (∃𝑦𝜑 → ∀𝑥∃𝑦𝜑)
4	2	hbeu 2098	. . 3 ⊢ (∃!𝑦𝜑 → ∀𝑥∃!𝑦𝜑)
5	3, 4	hbim 1591	. 2 ⊢ ((∃𝑦𝜑 → ∃!𝑦𝜑) → ∀𝑥(∃𝑦𝜑 → ∃!𝑦𝜑))
6	1, 5	hbxfrbi 1518	1 ⊢ (∃*𝑦𝜑 → ∀𝑥∃*𝑦𝜑)

Syntax hints:   → wi 4  ∀wal 1393  ∃wex 1538  ∃!weu 2077  ∃*wmo 2078

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081

This theorem is referenced by:  moexexdc  2162  2moex  2164  2euex  2165  2exeu  2170