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Mirrors > Home > ILE Home > Th. List > 3eqtr3ri | GIF version |
Description: An inference from three chained equalities. (Contributed by NM, 15-Aug-2004.) |
Ref | Expression |
---|---|
3eqtr3i.1 | ⊢ 𝐴 = 𝐵 |
3eqtr3i.2 | ⊢ 𝐴 = 𝐶 |
3eqtr3i.3 | ⊢ 𝐵 = 𝐷 |
Ref | Expression |
---|---|
3eqtr3ri | ⊢ 𝐷 = 𝐶 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 3eqtr3i.3 | . 2 ⊢ 𝐵 = 𝐷 | |
2 | 3eqtr3i.1 | . . 3 ⊢ 𝐴 = 𝐵 | |
3 | 3eqtr3i.2 | . . 3 ⊢ 𝐴 = 𝐶 | |
4 | 2, 3 | eqtr3i 2193 | . 2 ⊢ 𝐵 = 𝐶 |
5 | 1, 4 | eqtr3i 2193 | 1 ⊢ 𝐷 = 𝐶 |
Colors of variables: wff set class |
Syntax hints: = wceq 1348 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-5 1440 ax-gen 1442 ax-4 1503 ax-17 1519 ax-ext 2152 |
This theorem depends on definitions: df-bi 116 df-cleq 2163 |
This theorem is referenced by: indif2 3371 resdm2 5099 co01 5123 cocnvres 5133 undifdc 6897 1mhlfehlf 9083 rei 10850 resqrexlemover 10961 cos1bnd 11709 6gcd4e2 11937 3lcm2e6 12101 cosq23lt0 13469 sincos4thpi 13476 sincos6thpi 13478 cosq34lt1 13486 |
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