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

Step	Hyp	Ref	Expression
1		3eqtr4i.1	. . 3 ⊢ A:α
2		3eqtr4i.3	. . 3 ⊢ R⊧[S = A]
3	1, 2	eqtypri 81	. 2 ⊢ S:α
4		3eqtr4i.2	. . 3 ⊢ R⊧[A = B]
5	1, 4	eqtypi 78	. . . . 5 ⊢ B:α
6		3eqtr4i.4	. . . . 5 ⊢ R⊧[T = B]
7	5, 6	eqtypri 81	. . . 4 ⊢ T:α
8	7, 6	eqcomi 79	. . 3 ⊢ R⊧[B = T]
9	1, 4, 8	eqtri 95	. 2 ⊢ R⊧[A = T]
10	3, 2, 9	eqtri 95	1 ⊢ R⊧[S = T]

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