Theorem 3eqtr4i 86

Description: Transitivity of equality.

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 71	. 2 ⊢ S:α
4		3eqtr4i.2	. . 3 ⊢ R⊧[A = B]
5	1, 4	eqtypi 69	. . . . 5 ⊢ B:α
6		3eqtr4i.4	. . . . 5 ⊢ R⊧[T = B]
7	5, 6	eqtypri 71	. . . 4 ⊢ T:α
8	7, 6	eqcomi 70	. . 3 ⊢ R⊧[B = T]
9	1, 4, 8	eqtri 85	. 2 ⊢ R⊧[A = T]
10	3, 2, 9	eqtri 85	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-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