Theorem isfree 176

Description: Derive the metamath "is free" predicate in terms of the HOL "is free" predicate.

Hypotheses

Ref	Expression
alnex1.1	⊢ A:∗
isfree.2	⊢ ⊤⊧[(λx:α Ay:α) = A]

Assertion

Ref	Expression
isfree	⊢ ⊤⊧[A ⇒ (∀λx:α A)]

Distinct variable groups:   y,A   x,y,α

Proof of Theorem isfree

Step	Hyp	Ref	Expression
1		alnex1.1	. . . . 5 ⊢ A:∗
2	1	id 25	. . . 4 ⊢ A⊧A
3		isfree.2	. . . 4 ⊢ ⊤⊧[(λx:α Ay:α) = A]
4	2, 3	alrimi 170	. . 3 ⊢ A⊧(∀λx:α A)
5	3	ax-cb1 29	. . 3 ⊢ ⊤:∗
6	4, 5	adantl 51	. 2 ⊢ (⊤, A)⊧(∀λx:α A)
7	6	ex 148	1 ⊢ ⊤⊧[A ⇒ (∀λx:α A)]

Syntax hints:  tv 1  ∗hb 3  kc 5  λkl 6   = ke 7  ⊤kt 8  [kbr 9  ⊧wffMMJ2 11  wffMMJ2t 12   ⇒ tim 111  ∀tal 112

This theorem was proved from axioms:  ax-syl 15  ax-jca 17  ax-simpl 20  ax-simpr 21  ax-id 24  ax-trud 26  ax-cb1 29  ax-cb2 30  ax-refl 39  ax-eqmp 42  ax-ded 43  ax-ceq 46  ax-beta 60  ax-distrc 61  ax-leq 62  ax-distrl 63  ax-hbl1 93  ax-17 95  ax-inst 103  ax-eta 165

This theorem depends on definitions:  df-ov 65  df-al 116  df-an 118  df-im 119

This theorem is referenced by:  ax6  195  ax12  202  ax17  205