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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 2764 | . 2 ⊢ 𝐴 = 𝐶 |
4 | 3eqtr2i.3 | . 2 ⊢ 𝐶 = 𝐷 | |
5 | 3, 4 | eqtr2i 2762 | 1 ⊢ 𝐷 = 𝐴 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1542 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-9 2117 ax-ext 2704 |
This theorem depends on definitions: df-bi 206 df-an 398 df-ex 1783 df-cleq 2725 |
This theorem is referenced by: funimacnv 6586 uniqs 8722 ackbij1lem13 10176 ef01bndlem 16074 cos2bnd 16078 divalglem2 16285 decexp2 16955 lefld 18489 smndex2dlinvh 18735 discmp 22772 unmbl 24924 sinhalfpilem 25843 log2cnv 26317 lgam1 26436 ip0i 29816 polid2i 30148 hh0v 30159 pjinormii 30667 dfdec100 31782 dpmul100 31809 dpmul 31825 dpmul4 31826 subfacp1lem3 33840 uniqsALTV 36840 cotrclrcl 42106 sqwvfoura 44559 sqwvfourb 44560 |
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