Theorem hbmo1 2083

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 2049	. 2 ⊢ (∃*𝑥𝜑 ↔ (∃𝑥𝜑 → ∃!𝑥𝜑))
2		hbe1 1509	. . 3 ⊢ (∃𝑥𝜑 → ∀𝑥∃𝑥𝜑)
3		hbeu1 2055	. . 3 ⊢ (∃!𝑥𝜑 → ∀𝑥∃!𝑥𝜑)
4	2, 3	hbim 1559	. 2 ⊢ ((∃𝑥𝜑 → ∃!𝑥𝜑) → ∀𝑥(∃𝑥𝜑 → ∃!𝑥𝜑))
5	1, 4	hbxfrbi 1486	1 ⊢ (∃*𝑥𝜑 → ∀𝑥∃*𝑥𝜑)

Syntax hints:   → wi 4  ∀wal 1362  ∃wex 1506  ∃!weu 2045  ∃*wmo 2046

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-4 1524  ax-ial 1548  ax-i5r 1549

This theorem depends on definitions:  df-bi 117  df-eu 2048  df-mo 2049

This theorem is referenced by:  mopick2  2128  moexexdc  2129