Theorem hbmo 1033

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

Hypothesis

Ref	Expression
hbmo.1	⊢ (φ → ∀xφ)

Assertion

Ref	Expression
hbmo	⊢ (∃*yφ → ∀x∃*yφ)

Proof of Theorem hbmo

Step	Hyp	Ref	Expression
1		hbmo.1	. . . 4 ⊢ (φ → ∀xφ)
2	1	hbex 701	. . 3 ⊢ (∃yφ → ∀x∃yφ)
3	1	hbeu 1016	. . 3 ⊢ (∃!yφ → ∀x∃!yφ)
4	2, 3	hbim 702	. 2 ⊢ ((∃yφ → ∃!yφ) → ∀x(∃yφ → ∃!yφ))
5		df-mo 1010	. 2 ⊢ (∃*yφ ↔ (∃yφ → ∃!yφ))
6	5	bial 695	. 2 ⊢ (∀x∃*yφ ↔ ∀x(∃yφ → ∃!yφ))
7	4, 5, 6	3imtr4 192	1 ⊢ (∃*yφ → ∀x∃*yφ)

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

This theorem is referenced by:  moexex 1058  2moex 1060  2euex 1061  2exeu 1066  mosubop 1911  dffunmof 2678

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  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-17 925

This theorem depends on definitions:  df-bi 128  df-an 198  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010

metamath.org