Theorem isfree 188

Description: Derive the metamath "is free" predicate in terms of the HOL "is free" predicate. (Contributed by Mario Carneiro, 9-Oct-2014.)

Hypotheses

Ref	Expression
alnex1.1	|- A:*
isfree.2	|- T. |= [(\x:al Ay:al) = A]

Assertion

Ref	Expression
isfree	|- T. |= [A ==> (A.\x:al A)]

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

Proof of Theorem isfree

Step	Hyp	Ref	Expression
1		alnex1.1	. . . . 5 |- A:*
2	1	id 25	. . . 4 |- A |= A
3		isfree.2	. . . 4 |- T. |= [(\x:al Ay:al) = A]
4	2, 3	alrimi 182	. . 3 |- A |= (A.\x:al A)
5	3	ax-cb1 29	. . 3 |- T.:*
6	4, 5	adantl 56	. 2 |- (T., A) |= (A.\x:al A)
7	6	ex 158	1 |- T. |= [A ==> (A.\x:al A)]

Syntax hints:  tv 1  *hb 3  kc 5  \kl 6   = ke 7  T.kt 8  [kbr 9   |= wffMMJ2 11  wffMMJ2t 12   ==> tim 121  A.tal 122

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-wctl 31  ax-wctr 32  ax-weq 40  ax-refl 42  ax-eqmp 45  ax-ded 46  ax-wct 47  ax-wc 49  ax-ceq 51  ax-wv 63  ax-wl 65  ax-beta 67  ax-distrc 68  ax-leq 69  ax-distrl 70  ax-wov 71  ax-eqtypi 77  ax-eqtypri 80  ax-hbl1 103  ax-17 105  ax-inst 113  ax-eta 177

This theorem depends on definitions:  df-ov 73  df-al 126  df-an 128  df-im 129

This theorem is referenced by:  ax6  208  ax12  215  ax17m  218