Theorem hbmo1 2120

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 2086	. 2 ⊢ (∃*𝑥𝜑 ↔ (∃𝑥𝜑 → ∃!𝑥𝜑))
2		hbe1 1544	. . 3 ⊢ (∃𝑥𝜑 → ∀𝑥∃𝑥𝜑)
3		hbeu1 2092	. . 3 ⊢ (∃!𝑥𝜑 → ∀𝑥∃!𝑥𝜑)
4	2, 3	hbim 1594	. 2 ⊢ ((∃𝑥𝜑 → ∃!𝑥𝜑) → ∀𝑥(∃𝑥𝜑 → ∃!𝑥𝜑))
5	1, 4	hbxfrbi 1521	1 ⊢ (∃*𝑥𝜑 → ∀𝑥∃*𝑥𝜑)

Syntax hints:   → wi 4  ∀wal 1396  ∃wex 1541  ∃!weu 2082  ∃*wmo 2083

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-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-4 1559  ax-ial 1583  ax-i5r 1584

This theorem depends on definitions:  df-bi 117  df-eu 2085  df-mo 2086

This theorem is referenced by:  mopick2  2166  moexexdc  2167