Theorem hbmo1 1979

Description: Bound-variable hypothesis builder for "at most one." (Contributed by NM, 8-Mar-1995.)

Assertion

Ref	Expression
hbmo1	⊢ (∃*𝑥𝜑 → ∀𝑥∃*𝑥𝜑)

Proof of Theorem hbmo1

Step	Hyp	Ref	Expression
1		df-mo 1945	. 2 ⊢ (∃*𝑥𝜑 ↔ (∃𝑥𝜑 → ∃!𝑥𝜑))
2		hbe1 1424	. . 3 ⊢ (∃𝑥𝜑 → ∀𝑥∃𝑥𝜑)
3		hbeu1 1951	. . 3 ⊢ (∃!𝑥𝜑 → ∀𝑥∃!𝑥𝜑)
4	2, 3	hbim 1477	. 2 ⊢ ((∃𝑥𝜑 → ∃!𝑥𝜑) → ∀𝑥(∃𝑥𝜑 → ∃!𝑥𝜑))
5	1, 4	hbxfrbi 1401	1 ⊢ (∃*𝑥𝜑 → ∀𝑥∃*𝑥𝜑)

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-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-4 1440  ax-ial 1467  ax-i5r 1468

This theorem depends on definitions:  df-bi 115  df-eu 1944  df-mo 1945

This theorem is referenced by:  mopick2  2024  moexexdc  2025