Theorem hbmo 2065

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 2030	. 2 ⊢ (∃*𝑦𝜑 ↔ (∃𝑦𝜑 → ∃!𝑦𝜑))
2		hbmo.1	. . . 4 ⊢ (𝜑 → ∀𝑥𝜑)
3	2	hbex 1636	. . 3 ⊢ (∃𝑦𝜑 → ∀𝑥∃𝑦𝜑)
4	2	hbeu 2047	. . 3 ⊢ (∃!𝑦𝜑 → ∀𝑥∃!𝑦𝜑)
5	3, 4	hbim 1545	. 2 ⊢ ((∃𝑦𝜑 → ∃!𝑦𝜑) → ∀𝑥(∃𝑦𝜑 → ∃!𝑦𝜑))
6	1, 5	hbxfrbi 1472	1 ⊢ (∃*𝑦𝜑 → ∀𝑥∃*𝑦𝜑)

Syntax hints:   → wi 4  ∀wal 1351  ∃wex 1492  ∃!weu 2026  ∃*wmo 2027

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030

This theorem is referenced by:  moexexdc  2110  2moex  2112  2euex  2113  2exeu  2118