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Mirrors > Home > HOLE Home > Th. List > ax8 | GIF 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. |
Ref | Expression |
---|---|
ax8.1 | ⊢ A:α |
ax8.2 | ⊢ B:α |
ax8.3 | ⊢ C:α |
Ref | Expression |
---|---|
ax8 | ⊢ ⊤⊧[[A = B] ⇒ [[A = C] ⇒ [B = C]]] |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ax8.2 | . . . . 5 ⊢ B:α | |
2 | ax8.1 | . . . . . 6 ⊢ A:α | |
3 | 2, 1 | weqi 68 | . . . . . . 7 ⊢ [A = B]:∗ |
4 | ax8.3 | . . . . . . . 8 ⊢ C:α | |
5 | 2, 4 | weqi 68 | . . . . . . 7 ⊢ [A = C]:∗ |
6 | 3, 5 | simpl 22 | . . . . . 6 ⊢ ([A = B], [A = C])⊧[A = B] |
7 | 2, 6 | eqcomi 70 | . . . . 5 ⊢ ([A = B], [A = C])⊧[B = A] |
8 | 3, 5 | simpr 23 | . . . . 5 ⊢ ([A = B], [A = C])⊧[A = C] |
9 | 1, 7, 8 | eqtri 85 | . . . 4 ⊢ ([A = B], [A = C])⊧[B = C] |
10 | 9 | ex 148 | . . 3 ⊢ [A = B]⊧[[A = C] ⇒ [B = C]] |
11 | wtru 40 | . . 3 ⊢ ⊤:∗ | |
12 | 10, 11 | adantl 51 | . 2 ⊢ (⊤, [A = B])⊧[[A = C] ⇒ [B = C]] |
13 | 12 | ex 148 | 1 ⊢ ⊤⊧[[A = B] ⇒ [[A = C] ⇒ [B = C]]] |
Colors of variables: type var term |
Syntax hints: = ke 7 ⊤kt 8 [kbr 9 kct 10 ⊧wffMMJ2 11 wffMMJ2t 12 ⇒ tim 111 |
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-refl 39 ax-eqmp 42 ax-ded 43 ax-ceq 46 ax-beta 60 ax-distrc 61 ax-leq 62 ax-distrl 63 ax-hbl1 93 ax-17 95 ax-inst 103 |
This theorem depends on definitions: df-ov 65 df-an 118 df-im 119 |
This theorem is referenced by: (None) |
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