Theorem hbmo 2092

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 2057	. 2 ⊢ (∃*𝑦𝜑 ↔ (∃𝑦𝜑 → ∃!𝑦𝜑))
2		hbmo.1	. . . 4 ⊢ (𝜑 → ∀𝑥𝜑)
3	2	hbex 1658	. . 3 ⊢ (∃𝑦𝜑 → ∀𝑥∃𝑦𝜑)
4	2	hbeu 2074	. . 3 ⊢ (∃!𝑦𝜑 → ∀𝑥∃!𝑦𝜑)
5	3, 4	hbim 1567	. 2 ⊢ ((∃𝑦𝜑 → ∃!𝑦𝜑) → ∀𝑥(∃𝑦𝜑 → ∃!𝑦𝜑))
6	1, 5	hbxfrbi 1494	1 ⊢ (∃*𝑦𝜑 → ∀𝑥∃*𝑦𝜑)

Syntax hints:   → wi 4  ∀wal 1370  ∃wex 1514  ∃!weu 2053  ∃*wmo 2054

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 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557

This theorem depends on definitions:  df-bi 117  df-tru 1375  df-nf 1483  df-sb 1785  df-eu 2056  df-mo 2057

This theorem is referenced by:  moexexdc  2137  2moex  2139  2euex  2140  2exeu  2145