Theorem hbeu1 1015

Description: Bound-variable hypothesis builder for uniqueness.

Assertion

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

Proof of Theorem hbeu1

Step	Hyp	Ref	Expression
1		hba1 698	. . 3 ⊢ (∀x(φ ↔ x = y) → ∀x∀x(φ ↔ x = y))
2	1	hbex 701	. 2 ⊢ (∃y∀x(φ ↔ x = y) → ∀x∃y∀x(φ ↔ x = y))
3		df-eu 1009	. 2 ⊢ (∃!xφ ↔ ∃y∀x(φ ↔ x = y))
4	3	bial 695	. 2 ⊢ (∀x∃!xφ ↔ ∀x∃y∀x(φ ↔ x = y))
5	2, 3, 4	3imtr4 192	1 ⊢ (∃!xφ → ∀x∃!xφ)

Syntax hints:   → wi 2   ↔ wb 127  ∀wal 672  ∃wex 678   = weq 797  ∃!weu 1007

This theorem is referenced by:  hbmo1 1032  hbreu1 1307  dffun7 2688  fneu 2728  fv3 2839  tz6.12c 2846  aceq5lem5 3562

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677

This theorem depends on definitions:  df-bi 128  df-an 198  df-ex 679  df-eu 1009

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