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Theorem ecase 163
 Description: Elimination by cases. (Contributed by Mario Carneiro, 9-Oct-2014.)
Hypotheses
Ref Expression
ecase.1
ecase.2
ecase.3
ecase.4
ecase.5
ecase.6
Assertion
Ref Expression
ecase

Proof of Theorem ecase
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 ecase.3 . 2
2 ecase.6 . . 3
32ex 158 . 2
4 wim 137 . . . 4
5 ecase.2 . . . . 5
64, 5, 1wov 72 . . . 4
74, 6, 1wov 72 . . 3
8 ecase.5 . . . 4
98ex 158 . . 3
10 ecase.4 . . . . 5
1110ax-cb1 29 . . . . . 6
12 ecase.1 . . . . . . 7
1312, 5orval 147 . . . . . 6
1411, 13a1i 28 . . . . 5
1510, 14mpbi 82 . . . 4
16 wv 64 . . . . . . 7
174, 12, 16wov 72 . . . . . 6
184, 5, 16wov 72 . . . . . . 7
194, 18, 16wov 72 . . . . . 6
204, 17, 19wov 72 . . . . 5
2116, 1weqi 76 . . . . . . . 8
2221id 25 . . . . . . 7
234, 12, 16, 22oveq2 101 . . . . . 6
244, 5, 16, 22oveq2 101 . . . . . . 7
254, 18, 16, 24, 22oveq12 100 . . . . . 6
264, 17, 19, 23, 25oveq12 100 . . . . 5
2720, 1, 26cla4v 152 . . . 4
2815, 27syl 16 . . 3
297, 9, 28mpd 156 . 2
301, 3, 29mpd 156 1
 Colors of variables: type var term Syntax hints:  tv 1  hb 3  kc 5  kl 6   ke 7  kbr 9  kct 10   wffMMJ2 11  wffMMJ2t 12   tim 121  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
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