Theorem 3eqtr3i 87

Description: Transitivity of equality.

Hypotheses

Ref	Expression
3eqtr4i.1	⊢ A:α
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:α
2		3eqtr4i.2	. 2 ⊢ R⊧[A = B]
3		3eqtr3i.3	. . 3 ⊢ R⊧[A = S]
4	1, 3	eqcomi 70	. 2 ⊢ R⊧[S = A]
5	1, 2	eqtypi 69	. . 3 ⊢ B:α
6		3eqtr3i.4	. . 3 ⊢ R⊧[B = T]
7	5, 6	eqcomi 70	. 2 ⊢ R⊧[T = B]
8	1, 2, 4, 7	3eqtr4i 86	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:  dfan2  144  cbvf  167  leqf  169  axext  206