Theorem cla4v 152

Description: If A(x) is true for all x:al, then it is true for C = A(B). (Contributed by Mario Carneiro, 9-Oct-2014.)

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 66	. . 3 |- \x:al A:(al -> *)
3		cla4v.2	. . 3 |- B:al
4	2, 3	ax4g 149	. 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 116	. . 3 |- T. |= [(\x:al AB) = C]
8	5, 7	a1i 28	. 2 |- (A.\x:al A) |= [(\x:al AB) = C]
9	4, 8	mpbi 82	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 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-wov 71  ax-eqtypi 77  ax-eqtypri 80  ax-hbl1 103  ax-17 105  ax-inst 113

This theorem depends on definitions:  df-ov 73  df-al 126

This theorem is referenced by:  pm2.21  153  ecase  163  exlimdv2  166  ax4e  168  eta  178  exlimd  183  ac  197  ax10  213