Theorem hbeu1 2024

Description: Bound-variable hypothesis builder for uniqueness. (Contributed by NM, 9-Jul-1994.)

Assertion

Ref	Expression
hbeu1	⊢ (∃!𝑥𝜑 → ∀𝑥∃!𝑥𝜑)

Proof of Theorem hbeu1

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-eu 2017	. 2 ⊢ (∃!𝑥𝜑 ↔ ∃𝑦∀𝑥(𝜑 ↔ 𝑥 = 𝑦))
2		hba1 1528	. . 3 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑦) → ∀𝑥∀𝑥(𝜑 ↔ 𝑥 = 𝑦))
3	2	hbex 1624	. 2 ⊢ (∃𝑦∀𝑥(𝜑 ↔ 𝑥 = 𝑦) → ∀𝑥∃𝑦∀𝑥(𝜑 ↔ 𝑥 = 𝑦))
4	1, 3	hbxfrbi 1460	1 ⊢ (∃!𝑥𝜑 → ∀𝑥∃!𝑥𝜑)

Syntax hints:   → wi 4   ↔ wb 104  ∀wal 1341  ∃wex 1480  ∃!weu 2014

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-4 1498  ax-ial 1522

This theorem depends on definitions:  df-bi 116  df-eu 2017

This theorem is referenced by:  hbmo1  2052  eupicka  2094  exists2  2111