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Theorem 3eqtr4i 86
Description: Transitivity of equality.
Hypotheses
Ref Expression
3eqtr4i.1 |- A:al
3eqtr4i.2 |- R |= [A = B]
3eqtr4i.3 |- R |= [S = A]
3eqtr4i.4 |- R |= [T = B]
Assertion
Ref Expression
3eqtr4i |- R |= [S = T]
Proof of Theorem 3eqtr4i
StepHypRef Expression
1 3eqtr4i.1 . . 3 |- A:al
2 3eqtr4i.3 . . 3 |- R |= [S = A]
31, 2eqtypri 71 . 2 |- S:al
4 3eqtr4i.2 . . 3 |- R |= [A = B]
51, 4eqtypi 69 . . . . 5 |- B:al
6 3eqtr4i.4 . . . . 5 |- R |= [T = B]
75, 6eqtypri 71 . . . 4 |- T:al
87, 6eqcomi 70 . . 3 |- R |= [B = T]
91, 4, 8eqtri 85 . 2 |- R |= [A = T]
103, 2, 9eqtri 85 1 |- R |= [S = T]
Colors of variables: type var term
Syntax hints:   = ke 7  [kbr 9   |= wffMMJ2 11  wffMMJ2t 12
This theorem was proved from axioms:  ax-syl 15  ax-jca 17  ax-trud 26  ax-cb1 29  ax-cb2 30  ax-refl 39  ax-eqmp 42  ax-ceq 46
This theorem depends on definitions:  df-ov 65
This theorem is referenced by:  3eqtr3i  87  oveq123  88  hbxfrf  97  leqf  169  exnal  188
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