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Mirrors > Home > ILE Home > Th. List > 3eqtrri | GIF version |
Description: An inference from three chained equalities. (Contributed by NM, 3-Aug-2006.) (Proof shortened by Andrew Salmon, 25-May-2011.) |
Ref | Expression |
---|---|
3eqtri.1 | ⊢ 𝐴 = 𝐵 |
3eqtri.2 | ⊢ 𝐵 = 𝐶 |
3eqtri.3 | ⊢ 𝐶 = 𝐷 |
Ref | Expression |
---|---|
3eqtrri | ⊢ 𝐷 = 𝐴 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 3eqtri.1 | . . 3 ⊢ 𝐴 = 𝐵 | |
2 | 3eqtri.2 | . . 3 ⊢ 𝐵 = 𝐶 | |
3 | 1, 2 | eqtri 2160 | . 2 ⊢ 𝐴 = 𝐶 |
4 | 3eqtri.3 | . 2 ⊢ 𝐶 = 𝐷 | |
5 | 3, 4 | eqtr2i 2161 | 1 ⊢ 𝐷 = 𝐴 |
Colors of variables: wff set class |
Syntax hints: = wceq 1331 |
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 1423 ax-gen 1425 ax-4 1487 ax-17 1506 ax-ext 2121 |
This theorem depends on definitions: df-bi 116 df-cleq 2132 |
This theorem is referenced by: resindm 4861 dfdm2 5073 cofunex2g 6010 df1st2 6116 df2nd2 6117 enq0enq 7239 dfn2 8990 9p1e10 9184 0.999... 11290 sincosq3sgn 12909 sincosq4sgn 12910 |
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