Theorem ecase 163

Description: Elimination by cases. (Contributed by Mario Carneiro, 9-Oct-2014.)

Hypotheses

Ref	Expression
ecase.1	|- A:*
ecase.2	|- B:*
ecase.3	|- T:*
ecase.4	|- R |= [A \/ B]
ecase.5	|- (R, A) |= T
ecase.6	|- (R, B) |= T

Assertion

Ref	Expression
ecase	|- R |= T

Proof of Theorem ecase

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ecase.3	. 2 |- T:*
2		ecase.6	. . 3 |- (R, B) |= T
3	2	ex 158	. 2 |- R |= [B ==> T]
4		wim 137	. . . 4 |- ==> :(* -> (* -> *))
5		ecase.2	. . . . 5 |- B:*
6	4, 5, 1	wov 72	. . . 4 |- [B ==> T]:*
7	4, 6, 1	wov 72	. . 3 |- [[B ==> T] ==> T]:*
8		ecase.5	. . . 4 |- (R, A) |= T
9	8	ex 158	. . 3 |- R |= [A ==> T]
10		ecase.4	. . . . 5 |- R |= [A \/ B]
11	10	ax-cb1 29	. . . . . 6 |- R:*
12		ecase.1	. . . . . . 7 |- A:*
13	12, 5	orval 147	. . . . . 6 |- T. |= [[A \/ B] = (A.\x:* [[A ==> x:*] ==> [[B ==> x:*] ==> x:*]])]
14	11, 13	a1i 28	. . . . 5 |- R |= [[A \/ B] = (A.\x:* [[A ==> x:*] ==> [[B ==> x:*] ==> x:*]])]
15	10, 14	mpbi 82	. . . 4 |- R |= (A.\x:* [[A ==> x:*] ==> [[B ==> x:*] ==> x:*]])
16		wv 64	. . . . . . 7 |- x:*:*
17	4, 12, 16	wov 72	. . . . . 6 |- [A ==> x:*]:*
18	4, 5, 16	wov 72	. . . . . . 7 |- [B ==> x:*]:*
19	4, 18, 16	wov 72	. . . . . 6 |- [[B ==> x:*] ==> x:*]:*
20	4, 17, 19	wov 72	. . . . 5 |- [[A ==> x:*] ==> [[B ==> x:*] ==> x:*]]:*
21	16, 1	weqi 76	. . . . . . . 8 |- [x:* = T]:*
22	21	id 25	. . . . . . 7 |- [x:* = T] |= [x:* = T]
23	4, 12, 16, 22	oveq2 101	. . . . . 6 |- [x:* = T] |= [[A ==> x:*] = [A ==> T]]
24	4, 5, 16, 22	oveq2 101	. . . . . . 7 |- [x:* = T] |= [[B ==> x:*] = [B ==> T]]
25	4, 18, 16, 24, 22	oveq12 100	. . . . . 6 |- [x:* = T] |= [[[B ==> x:*] ==> x:*] = [[B ==> T] ==> T]]
26	4, 17, 19, 23, 25	oveq12 100	. . . . 5 |- [x:* = T] |= [[[A ==> x:*] ==> [[B ==> x:*] ==> x:*]] = [[A ==> T] ==> [[B ==> T] ==> T]]]
27	20, 1, 26	cla4v 152	. . . 4 |- (A.\x:* [[A ==> x:*] ==> [[B ==> x:*] ==> x:*]]) |= [[A ==> T] ==> [[B ==> T] ==> T]]
28	15, 27	syl 16	. . 3 |- R |= [[A ==> T] ==> [[B ==> T] ==> T]]
29	7, 9, 28	mpd 156	. 2 |- R |= [[B ==> T] ==> T]
30	1, 3, 29	mpd 156	1 |- R |= T

Syntax hints:  tv 1  *hb 3  kc 5  \kl 6   = ke 7  [kbr 9  kct 10   |= wffMMJ2 11  wffMMJ2t 12   ==> tim 121  A.tal 122   \/ tor 124

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-or 132

This theorem is referenced by:  exmid  199  notnot  200  ax3  205