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Mirrors > Home > HOLE Home > Th. List > 3eqtr4i | GIF version |
Description: Transitivity of equality. |
Ref | Expression |
---|---|
3eqtr4i.1 | ⊢ A:α |
3eqtr4i.2 | ⊢ R⊧[A = B] |
3eqtr4i.3 | ⊢ R⊧[S = A] |
3eqtr4i.4 | ⊢ R⊧[T = B] |
Ref | Expression |
---|---|
3eqtr4i | ⊢ R⊧[S = T] |
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] |
Colors of variables: type var term |
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 |
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