Theorem exlimdv 167

Description: Existential elimination. (Contributed by Mario Carneiro, 9-Oct-2014.)

Hypothesis

Ref	Expression
exlimdv.1	|- (R, A) |= T

Assertion

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

Distinct variable groups:   x,R   x,T   al,x

Proof of Theorem exlimdv

Dummy variables y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		exlimdv.1	. . . . 5 |- (R, A) |= T
2	1	ax-cb1 29	. . . 4 |- (R, A):*
3	2	wctr 34	. . 3 |- A:*
4	3	wl 66	. 2 |- \x:al A:(al -> *)
5		wv 64	. . . 4 |- y:al:al
6	4, 5	wc 50	. . 3 |- (\x:al Ay:al):*
7	1	ax-cb2 30	. . 3 |- T:*
8		wim 137	. . . . 5 |- ==> :(* -> (* -> *))
9	8, 6, 7	wov 72	. . . 4 |- [(\x:al Ay:al) ==> T]:*
10	1	ex 158	. . . 4 |- R |= [A ==> T]
11		wv 64	. . . . 5 |- z:al:al
12	8, 11	ax-17 105	. . . . 5 |- T. |= [(\x:al ==> z:al) = ==> ]
13	3, 11	ax-hbl1 103	. . . . . 6 |- T. |= [(\x:al \x:al Az:al) = \x:al A]
14	5, 11	ax-17 105	. . . . . 6 |- T. |= [(\x:al y:alz:al) = y:al]
15	4, 5, 11, 13, 14	hbc 110	. . . . 5 |- T. |= [(\x:al (\x:al Ay:al)z:al) = (\x:al Ay:al)]
16	7, 11	ax-17 105	. . . . 5 |- T. |= [(\x:al Tz:al) = T]
17	8, 6, 11, 7, 12, 15, 16	hbov 111	. . . 4 |- T. |= [(\x:al [(\x:al Ay:al) ==> T]z:al) = [(\x:al Ay:al) ==> T]]
18		wv 64	. . . . . . . 8 |- x:al:al
19	18, 5	weqi 76	. . . . . . 7 |- [x:al = y:al]:*
20	4, 18	wc 50	. . . . . . . 8 |- (\x:al Ax:al):*
21	3	beta 92	. . . . . . . 8 |- T. |= [(\x:al Ax:al) = A]
22	20, 21	eqcomi 79	. . . . . . 7 |- T. |= [A = (\x:al Ax:al)]
23	19, 22	a1i 28	. . . . . 6 |- [x:al = y:al] |= [A = (\x:al Ax:al)]
24	19	id 25	. . . . . . 7 |- [x:al = y:al] |= [x:al = y:al]
25	4, 18, 24	ceq2 90	. . . . . 6 |- [x:al = y:al] |= [(\x:al Ax:al) = (\x:al Ay:al)]
26	3, 23, 25	eqtri 95	. . . . 5 |- [x:al = y:al] |= [A = (\x:al Ay:al)]
27	8, 3, 7, 26	oveq1 99	. . . 4 |- [x:al = y:al] |= [[A ==> T] = [(\x:al Ay:al) ==> T]]
28	5, 9, 10, 17, 27	insti 114	. . 3 |- R |= [(\x:al Ay:al) ==> T]
29	6, 7, 28	imp 157	. 2 |- (R, (\x:al Ay:al)) |= T
30	4, 29	exlimdv2 166	1 |- (R, (E.\x:al A)) |= T

Syntax hints:  tv 1   -> ht 2  *hb 3  kc 5  \kl 6   = ke 7  T.kt 8  [kbr 9  kct 10   |= wffMMJ2 11   ==> tim 121  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)