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Mirrors > Home > ILE Home > Th. List > eleqtrd | Unicode version |
Description: Deduction that substitutes equal classes into membership. (Contributed by NM, 14-Dec-2004.) |
Ref | Expression |
---|---|
eleqtrd.1 | |
eleqtrd.2 |
Ref | Expression |
---|---|
eleqtrd |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eleqtrd.1 | . 2 | |
2 | eleqtrd.2 | . . 3 | |
3 | 2 | eleq2d 2187 | . 2 |
4 | 1, 3 | mpbid 146 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wceq 1316 wcel 1465 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-5 1408 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-4 1472 ax-17 1491 ax-ial 1499 ax-ext 2099 |
This theorem depends on definitions: df-bi 116 df-cleq 2110 df-clel 2113 |
This theorem is referenced by: eleqtrrd 2197 3eltr3d 2200 eleqtrid 2206 eleqtrdi 2210 opth1 4128 0nelop 4140 tfisi 4471 ercl 6408 erth 6441 ecelqsdm 6467 phpm 6727 suplocexprlemmu 7494 suplocexprlemloc 7497 lincmb01cmp 9754 fzopth 9809 fzoaddel2 9938 fzosubel2 9940 fzocatel 9944 zpnn0elfzo1 9953 fzoend 9967 peano2fzor 9977 monoord2 10218 ser3mono 10219 bcpasc 10480 zfz1isolemiso 10550 fisum0diag2 11184 isumsplit 11228 iscnp4 12314 cnrest2r 12333 txbasval 12363 txlm 12375 xmetunirn 12454 xblss2ps 12500 blbas 12529 mopntopon 12539 isxms2 12548 metcnpi 12611 metcnpi2 12612 tgioo 12642 cncfmpt2fcntop 12681 limccl 12724 limcimolemlt 12729 limccnp2cntop 12742 dvmulxxbr 12762 dvcoapbr 12767 dvcjbr 12768 dvrecap 12773 |
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