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Theorem 3eqtr4i 96
 Description: Transitivity of equality. (Contributed by Mario Carneiro, 7-Oct-2014.)
Hypotheses
Ref Expression
3eqtr4i.1 A:α
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:α
2 3eqtr4i.3 . . 3 R⊧[S = A]
31, 2eqtypri 81 . 2 S:α
4 3eqtr4i.2 . . 3 R⊧[A = B]
51, 4eqtypi 78 . . . . 5 B:α
6 3eqtr4i.4 . . . . 5 R⊧[T = B]
75, 6eqtypri 81 . . . 4 T:α
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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