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Mirrors > Home > MPE Home > Th. List > 3eqtr4ri | Structured version Visualization version GIF version |
Description: An inference from three chained equalities. (Contributed by NM, 2-Sep-1995.) (Proof shortened by Andrew Salmon, 25-May-2011.) |
Ref | Expression |
---|---|
3eqtr4i.1 | ⊢ 𝐴 = 𝐵 |
3eqtr4i.2 | ⊢ 𝐶 = 𝐴 |
3eqtr4i.3 | ⊢ 𝐷 = 𝐵 |
Ref | Expression |
---|---|
3eqtr4ri | ⊢ 𝐷 = 𝐶 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 3eqtr4i.3 | . . 3 ⊢ 𝐷 = 𝐵 | |
2 | 3eqtr4i.1 | . . 3 ⊢ 𝐴 = 𝐵 | |
3 | 1, 2 | eqtr4i 2635 | . 2 ⊢ 𝐷 = 𝐴 |
4 | 3eqtr4i.2 | . 2 ⊢ 𝐶 = 𝐴 | |
5 | 3, 4 | eqtr4i 2635 | 1 ⊢ 𝐷 = 𝐶 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1475 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-cleq 2603 |
This theorem is referenced by: cbvreucsf 3533 dfin3 3825 dfsymdif3 3852 dfif6 4039 qdass 4232 tpidm12 4234 opid 4359 unipr 4385 iinvdif 4528 unidif0 4764 csbcnv 5228 dfdm4 5238 dmun 5253 resres 5329 inres 5334 resdifcom 5335 resiun1 5336 resiun1OLD 5337 imainrect 5494 coundi 5553 coundir 5554 funopg 5836 csbov 6586 elrnmpt2res 6672 offres 7054 1st2val 7085 2nd2val 7086 mpt2mptsx 7122 oeoalem 7563 omopthlem1 7622 snec 7697 tcsni 8502 infmap2 8923 ackbij2lem3 8946 itunisuc 9124 axmulass 9857 divmul13i 10665 dfnn3 10911 halfpm6th 11130 numsucc 11425 decbin2 11559 uzrdgxfr 12628 hashxplem 13080 prprrab 13112 ids1 13230 s3s4 13528 s2s5 13529 s5s2 13530 fsumadd 14317 fsum2d 14344 fprodmul 14529 bpoly3 14628 bezout 15098 ressbas 15757 oppchomf 16203 oppr1 18457 opsrtoslem1 19305 m2detleiblem2 20253 cnfldnm 22392 cnnm 22768 volres 23103 voliunlem1 23125 uniioombllem4 23160 itg11 23264 plyid 23769 coeidp 23823 dgrid 23824 dfrelog 24116 log2ublem3 24475 bposlem8 24816 ip2i 27067 bcseqi 27361 hilnormi 27404 cmcmlem 27834 fh3i 27866 fh4i 27867 pjadjii 27917 cnvoprab 28886 resf1o 28893 ressplusf 28981 resvsca 29161 xpinpreima 29280 cnre2csqima 29285 esum2dlem 29481 eulerpartgbij 29761 ballotth 29926 plymulx 29951 elrn3 30906 domep 30942 itg2addnclem2 32632 areaquad 36821 cnvrcl0 36951 stoweidlem13 38906 wallispi2 38966 fourierdlem96 39095 fourierdlem97 39096 fourierdlem98 39097 fourierdlem99 39098 fourierdlem113 39112 fourierswlem 39123 fmtnorec2 39993 fmtno5fac 40032 uhgrspan1 40527 setrec2 42241 |
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