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| Mirrors > Home > MPE Home > Th. List > 3eqtr2ri | Structured version Visualization version 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 |
|---|---|
| 3eqtr2i.1 | ⊢ 𝐴 = 𝐵 |
| 3eqtr2i.2 | ⊢ 𝐶 = 𝐵 |
| 3eqtr2i.3 | ⊢ 𝐶 = 𝐷 |
| Ref | Expression |
|---|---|
| 3eqtr2ri | ⊢ 𝐷 = 𝐴 |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | 3eqtr2i.1 | . . 3 ⊢ 𝐴 = 𝐵 | |
| 2 | 3eqtr2i.2 | . . 3 ⊢ 𝐶 = 𝐵 | |
| 3 | 1, 2 | eqtr4i 2768 | . 2 ⊢ 𝐴 = 𝐶 |
| 4 | 3eqtr2i.3 | . 2 ⊢ 𝐶 = 𝐷 | |
| 5 | 3, 4 | eqtr2i 2766 | 1 ⊢ 𝐷 = 𝐴 |
| Colors of variables: wff setvar class |
| Syntax hints: = wceq 1540 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-9 2118 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1780 df-cleq 2729 |
| This theorem is referenced by: funimacnv 6647 uniqs 8817 ackbij1lem13 10271 ef01bndlem 16220 cos2bnd 16224 divalglem2 16432 lefld 18637 smndex2dlinvh 18930 discmp 23406 unmbl 25572 sinhalfpilem 26505 log2cnv 26987 lgam1 27107 ip0i 30844 polid2i 31176 hh0v 31187 pjinormii 31695 dfdec100 32832 dpmul100 32879 dpmul 32895 dpmul4 32896 subfacp1lem3 35187 uniqsALTV 38330 redvmptabs 42390 cotrclrcl 43755 sqwvfoura 46243 sqwvfourb 46244 |
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