Theorem 3eqtr3i 97

Description: Transitivity of equality. (Contributed by Mario Carneiro, 7-Oct-2014.)

Hypotheses

Ref	Expression
3eqtr4i.1	|- A:al
3eqtr4i.2	|- R |= [A = B]
3eqtr3i.3	|- R |= [A = S]
3eqtr3i.4	|- R |= [B = T]

Assertion

Ref	Expression
3eqtr3i	|- R |= [S = T]

Proof of Theorem 3eqtr3i

Step	Hyp	Ref	Expression
1		3eqtr4i.1	. 2 |- A:al
2		3eqtr4i.2	. 2 |- R |= [A = B]
3		3eqtr3i.3	. . 3 |- R |= [A = S]
4	1, 3	eqcomi 79	. 2 |- R |= [S = A]
5	1, 2	eqtypi 78	. . 3 |- B:al
6		3eqtr3i.4	. . 3 |- R |= [B = T]
7	5, 6	eqcomi 79	. 2 |- R |= [T = B]
8	1, 2, 4, 7	3eqtr4i 96	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:  dfan2  154  cbvf  179  leqf  181  axext  219