Theorem hbeu1 1386

Description: Bound-variable hypothesis builder for uniqueness.

Assertion

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

Proof of Theorem hbeu1

Step	Hyp	Ref	Expression
1		hba1 1001	. . 3 ⊢ (∀x(φ ↔ x = y) → ∀x∀x(φ ↔ x = y))
2	1	hbex 1004	. 2 ⊢ (∃y∀x(φ ↔ x = y) → ∀x∃y∀x(φ ↔ x = y))
3		df-eu 1380	. 2 ⊢ (∃!xφ ↔ ∃y∀x(φ ↔ x = y))
4	3	albii 997	. 2 ⊢ (∀x∃!xφ ↔ ∀x∃y∀x(φ ↔ x = y))
5	2, 3, 4	3imtr4 219	1 ⊢ (∃!xφ → ∀x∃!xφ)

Syntax hints:   → wi 3   ↔ wb 146  ∀wal 952  ∃wex 978  ∃!weu 1378

This theorem is referenced by:  hbmo1 1404  moaneu 1428  2eu8 1454  hbreu1 1765  dffun7 3532  fneu 3584  fv3 3724  tz6.12c 3731  aceq5lem5 4719

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-4 971  ax-5o 973  ax-6o 976

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 979  df-eu 1380

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