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Mirrors > Home > HOLE Home > Th. List > ax9 | Unicode version |
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.) |
Ref | Expression |
---|---|
ax9.1 |
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Ref | Expression |
---|---|
ax9 |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | wv 64 |
. . . . . 6
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2 | ax9.1 |
. . . . . 6
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3 | 1, 2 | weqi 76 |
. . . . 5
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4 | 3 | 19.8a 170 |
. . . 4
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5 | wex 139 |
. . . . 5
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6 | 3 | wl 66 |
. . . . 5
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7 | wv 64 |
. . . . 5
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8 | 5, 7 | ax-17 105 |
. . . . 5
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9 | 3, 7 | ax-hbl1 103 |
. . . . 5
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10 | 5, 6, 7, 8, 9 | hbc 110 |
. . . 4
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11 | wtru 43 |
. . . . 5
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12 | 11, 7 | ax-17 105 |
. . . 4
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13 | 5, 6 | wc 50 |
. . . . 5
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14 | 3, 13 | eqid 83 |
. . . 4
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15 | 3 | id 25 |
. . . . . 6
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16 | 15 | eqtru 86 |
. . . . 5
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17 | 11, 16 | eqcomi 79 |
. . . 4
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18 | 4, 10, 12, 14, 17 | ax-inst 113 |
. . 3
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19 | 13 | notnot1 160 |
. . 3
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20 | 18, 19 | syl 16 |
. 2
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21 | wnot 138 |
. . 3
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22 | wal 134 |
. . . 4
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23 | 21, 3 | wc 50 |
. . . . 5
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24 | 23 | wl 66 |
. . . 4
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25 | 22, 24 | wc 50 |
. . 3
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26 | 3 | alnex 186 |
. . 3
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27 | 21, 25, 26 | ceq2 90 |
. 2
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28 | 20, 27 | mpbir 87 |
1
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Colors of variables: type var term |
Syntax hints: tv 1
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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) |
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