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Mirrors > Home > MPE Home > Th. List > 3eqtr3a | Structured version Visualization version GIF version |
Description: A chained equality inference, useful for converting from definitions. (Contributed by Mario Carneiro, 6-Nov-2015.) |
Ref | Expression |
---|---|
3eqtr3a.1 | ⊢ 𝐴 = 𝐵 |
3eqtr3a.2 | ⊢ (𝜑 → 𝐴 = 𝐶) |
3eqtr3a.3 | ⊢ (𝜑 → 𝐵 = 𝐷) |
Ref | Expression |
---|---|
3eqtr3a | ⊢ (𝜑 → 𝐶 = 𝐷) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 3eqtr3a.2 | . 2 ⊢ (𝜑 → 𝐴 = 𝐶) | |
2 | 3eqtr3a.1 | . . 3 ⊢ 𝐴 = 𝐵 | |
3 | 3eqtr3a.3 | . . 3 ⊢ (𝜑 → 𝐵 = 𝐷) | |
4 | 2, 3 | eqtrid 2785 | . 2 ⊢ (𝜑 → 𝐴 = 𝐷) |
5 | 1, 4 | eqtr3d 2775 | 1 ⊢ (𝜑 → 𝐶 = 𝐷) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = 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: uneqin 4279 coi2 6263 foima 6811 f1imacnv 6850 fvsnun1 7180 fnsnsplit 7182 phplem2 9208 php3 9212 phplem4OLD 9220 php3OLD 9224 rankopb 9847 fin4en1 10304 fpwwe2 10638 winacard 10687 mul02lem1 11390 cnegex2 11396 crreczi 14191 hashinf 14295 hashcard 14315 cshw0 14744 cshwn 14747 sqrtneglem 15213 rlimresb 15509 bpoly3 16002 bpoly4 16003 sinhval 16097 coshval 16098 absefib 16141 efieq1re 16142 sadcaddlem 16398 sadaddlem 16407 qus0subgbas 19075 psgnsn 19388 odngen 19445 frlmup3 21355 mat0op 21921 restopnb 22679 cnmpt2t 23177 clmnegneg 24620 ncvspi 24673 volsup2 25122 plypf1 25726 pige3ALT 26029 sineq0 26033 eflog 26085 logef 26090 cxpsqrt 26211 dvcncxp1 26251 cubic2 26353 quart1 26361 asinsinlem 26396 asinsin 26397 2efiatan 26423 pclogsum 26718 lgsneg 26824 vc0 29858 vcm 29860 nvpi 29951 honegneg 31090 opsqrlem6 31429 sto1i 31520 mdexchi 31619 preiman0 31962 elrspunidl 32577 cnre2csqlem 32921 itgexpif 33649 subfacp1lem1 34201 rankaltopb 34982 poimirlem23 36559 dvtan 36586 dvasin 36620 heiborlem6 36732 trlcoat 39642 cdlemk54 39877 resubid 41329 sn-mul02 41361 iocunico 42008 relintab 42382 rfovcnvf1od 42803 ntrneifv3 42881 ntrneifv4 42884 clsneifv3 42909 clsneifv4 42910 neicvgfv 42920 snunioo1 44273 dvsinexp 44675 itgsubsticclem 44739 stirlinglem1 44838 fourierdlem80 44950 fourierdlem111 44981 sqwvfoura 44992 sqwvfourb 44993 fouriersw 44995 saliinclf 45090 smfco 45566 aacllem 47896 |
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