Theorem eximdv 185

Description: Deduction from Theorem 19.22 of [Margaris] p. 90. (Contributed by Mario Carneiro, 9-Oct-2014.)

Hypothesis

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

Assertion

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

Distinct variable groups:   x,R   al,x

Proof of Theorem eximdv

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		alimdv.1	. . 3 |- (R, A) |= B
2	1	ax-cb2 30	. . . 4 |- B:*
3	2	19.8a 170	. . 3 |- B |= (E.\x:al B)
4	1, 3	syl 16	. 2 |- (R, A) |= (E.\x:al B)
5	1	ax-cb1 29	. . . 4 |- (R, A):*
6	5	wctl 33	. . 3 |- R:*
7		wv 64	. . 3 |- y:al:al
8	6, 7	ax-17 105	. 2 |- T. |= [(\x:al Ry:al) = R]
9		wex 139	. . 3 |- E.:((al -> *) -> *)
10	2	wl 66	. . 3 |- \x:al B:(al -> *)
11	9, 7	ax-17 105	. . 3 |- T. |= [(\x:al E.y:al) = E.]
12	2, 7	ax-hbl1 103	. . 3 |- T. |= [(\x:al \x:al By:al) = \x:al B]
13	9, 10, 7, 11, 12	hbc 110	. 2 |- T. |= [(\x:al (E.\x:al B)y:al) = (E.\x:al B)]
14	4, 8, 13	exlimd 183	1 |- (R, (E.\x:al A)) |= (E.\x:al B)

Syntax hints:  tv 1   -> ht 2  *hb 3  kc 5  \kl 6  T.kt 8  kct 10   |= wffMMJ2 11  E.tex 123

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

This theorem depends on definitions:  df-ov 73  df-al 126  df-an 128  df-im 129  df-ex 131

This theorem is referenced by: (None)