Theorem 3eqtr4i 86

Description: Transitivity of equality.

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 71	. 2 |- S:al
4		3eqtr4i.2	. . 3 |- R |= [A = B]
5	1, 4	eqtypi 69	. . . . 5 |- B:al
6		3eqtr4i.4	. . . . 5 |- R |= [T = B]
7	5, 6	eqtypri 71	. . . 4 |- T:al
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