Higher-Order Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  HOLE Home  >  Th. List  >  ax9 Unicode version

Theorem ax9 212
 Description: Axiom of Equality. Axiom scheme C8' in [Megill] p. 448 (p. 16 of the preprint). Also appears as Axiom C7 of [Monk2] p. 105. (Contributed by Mario Carneiro, 10-Oct-2014.)
Hypothesis
Ref Expression
ax9.1
Assertion
Ref Expression
ax9
Distinct variable group:   ,

Proof of Theorem ax9
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 wv 64 . . . . . 6
2 ax9.1 . . . . . 6
31, 2weqi 76 . . . . 5
4319.8a 170 . . . 4
5 wex 139 . . . . 5
63wl 66 . . . . 5
7 wv 64 . . . . 5
85, 7ax-17 105 . . . . 5
93, 7ax-hbl1 103 . . . . 5
105, 6, 7, 8, 9hbc 110 . . . 4
11 wtru 43 . . . . 5
1211, 7ax-17 105 . . . 4
135, 6wc 50 . . . . 5
143, 13eqid 83 . . . 4
153id 25 . . . . . 6
1615eqtru 86 . . . . 5
1711, 16eqcomi 79 . . . 4
184, 10, 12, 14, 17ax-inst 113 . . 3
1913notnot1 160 . . 3
2018, 19syl 16 . 2
21 wnot 138 . . 3
22 wal 134 . . . 4
2321, 3wc 50 . . . . 5
2423wl 66 . . . 4
2522, 24wc 50 . . 3
263alnex 186 . . 3
2721, 25, 26ceq2 90 . 2
2820, 27mpbir 87 1
 Colors of variables: type var term Syntax hints:  tv 1   ht 2  hb 3  kc 5  kl 6   ke 7  kt 8  kbr 9   wffMMJ2 11  wffMMJ2t 12   tne 120  tal 122  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  ax-eta 177 This theorem depends on definitions:  df-ov 73  df-al 126  df-fal 127  df-an 128  df-im 129  df-not 130  df-ex 131 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator