Theorem hbeu1 1951

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 1944	. 2 ⊢ (∃!𝑥𝜑 ↔ ∃𝑦∀𝑥(𝜑 ↔ 𝑥 = 𝑦))
2		hba1 1473	. . 3 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑦) → ∀𝑥∀𝑥(𝜑 ↔ 𝑥 = 𝑦))
3	2	hbex 1567	. 2 ⊢ (∃𝑦∀𝑥(𝜑 ↔ 𝑥 = 𝑦) → ∀𝑥∃𝑦∀𝑥(𝜑 ↔ 𝑥 = 𝑦))
4	1, 3	hbxfrbi 1401	1 ⊢ (∃!𝑥𝜑 → ∀𝑥∃!𝑥𝜑)

Syntax hints:   → wi 4   ↔ wb 103  ∀wal 1282  ∃wex 1421  ∃!weu 1941

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

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

This theorem is referenced by:  hbmo1  1979  eupicka  2021  exists2  2038