Theorem hbmo1 1032

Description: Bound-variable hypothesis builder for "at most one".

Assertion

Ref	Expression
hbmo1	⊢ (∃*xφ → ∀x∃*xφ)

Proof of Theorem hbmo1

Step	Hyp	Ref	Expression
1		hbe1 709	. . 3 ⊢ (∃xφ → ∀x∃xφ)
2		hbeu1 1015	. . 3 ⊢ (∃!xφ → ∀x∃!xφ)
3	1, 2	hbim 702	. 2 ⊢ ((∃xφ → ∃!xφ) → ∀x(∃xφ → ∃!xφ))
4		df-mo 1010	. 2 ⊢ (∃*xφ ↔ (∃xφ → ∃!xφ))
5	4	bial 695	. 2 ⊢ (∀x∃*xφ ↔ ∀x(∃xφ → ∃!xφ))
6	3, 4, 5	3imtr4 192	1 ⊢ (∃*xφ → ∀x∃*xφ)

Syntax hints:   → wi 2  ∀wal 672  ∃wex 678  ∃!weu 1007  ∃*wmo 1008

This theorem is referenced by:  mopick2 1057  moexex 1058  2eu3 1069

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  df-mo 1010

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