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Theorem ax8 198
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.
Hypotheses
Ref Expression
ax8.1 |- A:al
ax8.2 |- B:al
ax8.3 |- C:al
Assertion
Ref Expression
ax8 |- T. |= [[A = B] ==> [[A = C] ==> [B = C]]]
Proof of Theorem ax8
StepHypRef Expression
1 ax8.2 . . . . 5 |- B:al
2 ax8.1 . . . . . 6 |- A:al
32, 1weqi 68 . . . . . . 7 |- [A = B]:*
4 ax8.3 . . . . . . . 8 |- C:al
52, 4weqi 68 . . . . . . 7 |- [A = C]:*
63, 5simpl 22 . . . . . 6 |- ([A = B], [A = C]) |= [A = B]
72, 6eqcomi 70 . . . . 5 |- ([A = B], [A = C]) |= [B = A]
83, 5simpr 23 . . . . 5 |- ([A = B], [A = C]) |= [A = C]
91, 7, 8eqtri 85 . . . 4 |- ([A = B], [A = C]) |= [B = C]
109ex 148 . . 3 |- [A = B] |= [[A = C] ==> [B = C]]
11 wtru 40 . . 3 |- T.:*
1210, 11adantl 51 . 2 |- (T., [A = B]) |= [[A = C] ==> [B = C]]
1312ex 148 1 |- T. |= [[A = B] ==> [[A = C] ==> [B = C]]]
Colors of variables: type var term
Syntax hints:   = ke 7  T.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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