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

Step	Hyp	Ref	Expression
1		3eqtr4i.1	. . 3 |- A:al
2		3eqtr4i.3	. . 3 |- R |= [S = A]
3	1, 2	eqtypri 81	. 2 |- S:al
4		3eqtr4i.2	. . 3 |- R |= [A = B]
5	1, 4	eqtypi 78	. . . . 5 |- B:al
6		3eqtr4i.4	. . . . 5 |- R |= [T = B]
7	5, 6	eqtypri 81	. . . 4 |- T:al
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