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Theorem 3eqtr4i 96
Description: Transitivity of equality. (Contributed by Mario Carneiro, 7-Oct-2014.)
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 81 . 2 |- S:al
4 3eqtr4i.2 . . 3 |- R |= [A = B]
51, 4eqtypi 78 . . . . 5 |- B:al
6 3eqtr4i.4 . . . . 5 |- R |= [T = B]
75, 6eqtypri 81 . . . 4 |- T:al
87, 6eqcomi 79 . . 3 |- R |= [B = T]
91, 4, 8eqtri 95 . 2 |- R |= [A = T]
103, 2, 9eqtri 95 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-weq 40  ax-refl 42  ax-eqmp 45  ax-wc 49  ax-ceq 51  ax-wov 71  ax-eqtypi 77  ax-eqtypri 80
This theorem depends on definitions:  df-ov 73
This theorem is referenced by:  3eqtr3i  97  oveq123  98  hbxfrf  107  leqf  181  exnal  201
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