Theorem cla4v 142

Description: If A(x) is true for all x:al, then it is true for C = A(B).

Hypotheses

Ref	Expression
cla4v.1	|- A:*
cla4v.2	|- B:al
cla4v.3	|- [x:al = B] |= [A = C]

Assertion

Ref	Expression
cla4v	|- (A.\x:al A) |= C

Distinct variable groups:   x,B   x,C   al,x

Proof of Theorem cla4v

Step	Hyp	Ref	Expression
1		cla4v.1	. . . 4 |- A:*
2	1	wl 59	. . 3 |- \x:al A:(al -> *)
3		cla4v.2	. . 3 |- B:al
4	2, 3	ax4g 139	. 2 |- (A.\x:al A) |= (\x:al AB)
5	4	ax-cb1 29	. . 3 |- (A.\x:al A):*
6		cla4v.3	. . . 4 |- [x:al = B] |= [A = C]
7	1, 3, 6	cl 106	. . 3 |- T. |= [(\x:al AB) = C]
8	5, 7	a1i 28	. 2 |- (A.\x:al A) |= [(\x:al AB) = C]
9	4, 8	mpbi 72	1 |- (A.\x:al A) |= C

Syntax hints:  tv 1  *hb 3  kc 5  \kl 6   = ke 7  [kbr 9   |= wffMMJ2 11  wffMMJ2t 12  A.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-hbl1 93  ax-17 95  ax-inst 103

This theorem depends on definitions:  df-ov 65  df-al 116

This theorem is referenced by:  pm2.21  143  ecase  153  exlimdv2  156  ax4e  158  eta  166  exlimd  171  ac  184  ax10  200