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Mirrors > Home > MPE Home > Th. List > eqnetri | Structured version Visualization version GIF version |
Description: Substitution of equal classes into an inequality. (Contributed by NM, 4-Jul-2012.) |
Ref | Expression |
---|---|
eqnetr.1 | ⊢ 𝐴 = 𝐵 |
eqnetr.2 | ⊢ 𝐵 ≠ 𝐶 |
Ref | Expression |
---|---|
eqnetri | ⊢ 𝐴 ≠ 𝐶 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqnetr.2 | . 2 ⊢ 𝐵 ≠ 𝐶 | |
2 | eqnetr.1 | . . 3 ⊢ 𝐴 = 𝐵 | |
3 | 2 | neeq1i 3002 | . 2 ⊢ (𝐴 ≠ 𝐶 ↔ 𝐵 ≠ 𝐶) |
4 | 1, 3 | mpbir 231 | 1 ⊢ 𝐴 ≠ 𝐶 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1536 ≠ wne 2937 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-9 2115 ax-ext 2705 |
This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1776 df-cleq 2726 df-ne 2938 |
This theorem is referenced by: eqnetrri 3009 notzfaus 5368 2on0 8520 1n0 8524 snnen2o 9270 noinfep 9697 card1 10005 fin23lem31 10380 s1nz 14641 bpoly4 16091 tan0 16183 nn0rppwr 16594 resslemOLD 17287 basendxnplusgndxOLD 17328 basendxnmulrndx 17340 basendxnmulrndxOLD 17341 plusgndxnmulrndx 17342 slotsbhcdif 17460 slotsbhcdifOLD 17461 symgvalstructOLD 19429 rmodislmodOLD 20945 cnfldfunALTOLDOLD 21410 xrsnsgrp 21437 pzriprnglem4 21512 matvscaOLD 22437 ustuqtop1 24265 iaa 26381 tan4thpi 26570 tan4thpiOLD 26571 ang180lem2 26867 mcubic 26904 quart1lem 26912 nogt01o 27755 slotsinbpsd 28463 slotslnbpsd 28464 ex-lcm 30486 9p10ne21 30498 resvlemOLD 33337 esumnul 34028 ballotth 34518 quad3 35654 bj-1upln0 36991 bj-2upln0 37005 bj-2upln1upl 37006 tan3rdpi 42364 sn-0ne2 42412 flt0 42623 flt4lem5e 42642 mncn0 43127 aaitgo 43150 mnringnmulrdOLD 44205 stirlinglem11 46039 sec0 48990 2p2ne5 49028 |
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