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Mirrors > Home > MPE Home > Th. List > oveqdr | Structured version Visualization version GIF version |
Description: Equality of two operations for any two operands. Useful in proofs using *propd theorems. (Contributed by Mario Carneiro, 29-Jun-2015.) |
Ref | Expression |
---|---|
oveqdr.1 | ⊢ (𝜑 → 𝐹 = 𝐺) |
Ref | Expression |
---|---|
oveqdr | ⊢ ((𝜑 ∧ 𝜓) → (𝑥𝐹𝑦) = (𝑥𝐺𝑦)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | oveqdr.1 | . . 3 ⊢ (𝜑 → 𝐹 = 𝐺) | |
2 | 1 | oveqd 7447 | . 2 ⊢ (𝜑 → (𝑥𝐹𝑦) = (𝑥𝐺𝑦)) |
3 | 2 | adantr 480 | 1 ⊢ ((𝜑 ∧ 𝜓) → (𝑥𝐹𝑦) = (𝑥𝐺𝑦)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1536 (class class class)co 7430 |
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-8 2107 ax-9 2115 ax-ext 2705 |
This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1539 df-ex 1776 df-sb 2062 df-clab 2712 df-cleq 2726 df-clel 2813 df-v 3479 df-ss 3979 df-uni 4912 df-br 5148 df-iota 6515 df-fv 6570 df-ov 7433 |
This theorem is referenced by: fullresc 17901 fucpropd 18033 resssetc 18145 resscatc 18162 issstrmgm 18678 gsumpropd 18703 issubmgm2 18728 grpsubpropd 19075 sylow2blem2 19653 isrngd 20190 prdsrngd 20193 isringd 20304 prdsringd 20334 prdscrngd 20335 prds1 20336 rnghmval 20456 pwsco1rhm 20518 pwsco2rhm 20519 pwsdiagrhm 20623 rnghmsubcsetclem1 20647 rnghmsubcsetclem2 20648 rngcifuestrc 20655 rhmsubcsetclem1 20676 rhmsubcsetclem2 20677 rhmsubcrngclem1 20682 rhmsubcrngclem2 20683 isdomn 20721 primefld 20822 sraring 21210 sralmod 21211 sralmod0 21212 issubrgd 21213 znzrh 21578 zncrng 21580 phlssphl 21694 sraassaOLD 21907 opsrcrng 22100 opsrassa 22101 ply1lss 22213 ply1subrg 22214 opsr0 22235 opsr1 22236 subrgply1 22249 opsrring 22261 opsrlmod 22262 ply1mpl0 22273 ply1mpl1 22275 ply1ascl 22276 coe1tm 22291 evls1rhm 22341 evl1rhm 22351 evl1expd 22364 evls1maplmhm 22396 mat0 22438 matinvg 22439 matlmod 22450 scmatsrng1 22544 1mavmul 22569 mat2pmatmul 22752 ressprdsds 24396 nmpropd 24622 tng0 24674 tngngp2 24688 tnggrpr 24691 tngnrg 24710 sranlm 24720 pi1addval 25094 cvsi 25176 tcphphl 25274 abvpropd2 32934 resv0g 33346 resvcmn 33348 sra1r 33611 sradrng 33612 srasubrg 33613 drgextlsp 33622 tnglvec 33639 tngdim 33640 matdim 33642 fedgmullem2 33657 zhmnrg 33927 prdsbnd 37779 prdstotbnd 37780 prdsbnd2 37781 erngdvlem3 40972 erngdvlem3-rN 40980 hlhils0 41931 hlhils1N 41932 hlhillvec 41937 hlhildrng 41938 hlhil0 41941 hlhillsm 41942 zndvdchrrhm 41952 isprimroot 42074 primrootsunit1 42078 mendval 43167 mnring0gd 44214 mnringlmodd 44221 |
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