Theorem hbmo 1980

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 1945	. 2 ⊢ (∃*𝑦𝜑 ↔ (∃𝑦𝜑 → ∃!𝑦𝜑))
2		hbmo.1	. . . 4 ⊢ (𝜑 → ∀𝑥𝜑)
3	2	hbex 1567	. . 3 ⊢ (∃𝑦𝜑 → ∀𝑥∃𝑦𝜑)
4	2	hbeu 1962	. . 3 ⊢ (∃!𝑦𝜑 → ∀𝑥∃!𝑦𝜑)
5	3, 4	hbim 1477	. 2 ⊢ ((∃𝑦𝜑 → ∃!𝑦𝜑) → ∀𝑥(∃𝑦𝜑 → ∃!𝑦𝜑))
6	1, 5	hbxfrbi 1401	1 ⊢ (∃*𝑦𝜑 → ∀𝑥∃*𝑦𝜑)

Syntax hints:   → wi 4  ∀wal 1282  ∃wex 1421  ∃!weu 1941  ∃*wmo 1942

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945

This theorem is referenced by:  moexexdc  2025  2moex  2027  2euex  2028  2exeu  2033