Theorem alimdv 184

Description: Deduction from Theorem 19.20 of [Margaris] p. 90. (Contributed by Mario Carneiro, 9-Oct-2014.)

Hypothesis

Ref	Expression
alimdv.1	|- (R, A) |= B

Assertion

Ref	Expression
alimdv	|- (R, (A.\x:al A)) |= (A.\x:al B)

Distinct variable groups:   x,R   al,x

Proof of Theorem alimdv

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		alimdv.1	. . . . . . 7 |- (R, A) |= B
2	1	ax-cb1 29	. . . . . 6 |- (R, A):*
3	2	wctr 34	. . . . 5 |- A:*
4	3	ax4 150	. . . 4 |- (A.\x:al A) |= A
5	2	wctl 33	. . . 4 |- R:*
6	4, 5	adantl 56	. . 3 |- (R, (A.\x:al A)) |= A
7	6, 1	syldan 36	. 2 |- (R, (A.\x:al A)) |= B
8		wv 64	. . 3 |- y:al:al
9		wal 134	. . . 4 |- A.:((al -> *) -> *)
10	3	wl 66	. . . 4 |- \x:al A:(al -> *)
11	9, 10	wc 50	. . 3 |- (A.\x:al A):*
12	5, 8	ax-17 105	. . 3 |- T. |= [(\x:al Ry:al) = R]
13	9, 8	ax-17 105	. . . 4 |- T. |= [(\x:al A.y:al) = A.]
14	3, 8	ax-hbl1 103	. . . 4 |- T. |= [(\x:al \x:al Ay:al) = \x:al A]
15	9, 10, 8, 13, 14	hbc 110	. . 3 |- T. |= [(\x:al (A.\x:al A)y:al) = (A.\x:al A)]
16	5, 8, 11, 12, 15	hbct 155	. 2 |- T. |= [(\x:al (R, (A.\x:al A))y:al) = (R, (A.\x:al A))]
17	7, 16	alrimi 182	1 |- (R, (A.\x:al A)) |= (A.\x:al B)

Syntax hints:  tv 1   -> ht 2  *hb 3  kc 5  \kl 6  T.kt 8  kct 10   |= wffMMJ2 11  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

This theorem is referenced by:  exnal1  187