Theorem hbmo1 1980

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 1946	. 2 ⊢ (∃*𝑥𝜑 ↔ (∃𝑥𝜑 → ∃!𝑥𝜑))
2		hbe1 1425	. . 3 ⊢ (∃𝑥𝜑 → ∀𝑥∃𝑥𝜑)
3		hbeu1 1952	. . 3 ⊢ (∃!𝑥𝜑 → ∀𝑥∃!𝑥𝜑)
4	2, 3	hbim 1478	. 2 ⊢ ((∃𝑥𝜑 → ∃!𝑥𝜑) → ∀𝑥(∃𝑥𝜑 → ∃!𝑥𝜑))
5	1, 4	hbxfrbi 1402	1 ⊢ (∃*𝑥𝜑 → ∀𝑥∃*𝑥𝜑)

Syntax hints:   → wi 4  ∀wal 1283  ∃wex 1422  ∃!weu 1942  ∃*wmo 1943

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 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-4 1441  ax-ial 1468  ax-i5r 1469

This theorem depends on definitions:  df-bi 115  df-eu 1945  df-mo 1946

This theorem is referenced by:  mopick2  2025  moexexdc  2026