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Mirrors > Home > ILE Home > Th. List > oveq12 | GIF version |
Description: Equality theorem for operation value. (Contributed by NM, 16-Jul-1995.) |
Ref | Expression |
---|---|
oveq12 | ⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → (𝐴𝐹𝐶) = (𝐵𝐹𝐷)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | oveq1 5857 | . 2 ⊢ (𝐴 = 𝐵 → (𝐴𝐹𝐶) = (𝐵𝐹𝐶)) | |
2 | oveq2 5858 | . 2 ⊢ (𝐶 = 𝐷 → (𝐵𝐹𝐶) = (𝐵𝐹𝐷)) | |
3 | 1, 2 | sylan9eq 2223 | 1 ⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → (𝐴𝐹𝐶) = (𝐵𝐹𝐷)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 = wceq 1348 (class class class)co 5850 |
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-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-ext 2152 |
This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-nf 1454 df-sb 1756 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-rex 2454 df-v 2732 df-un 3125 df-sn 3587 df-pr 3588 df-op 3590 df-uni 3795 df-br 3988 df-iota 5158 df-fv 5204 df-ov 5853 |
This theorem is referenced by: oveq12i 5862 oveq12d 5868 oveqan12d 5869 ecopoveq 6604 ecopovtrn 6606 ecopovtrng 6609 th3qlem1 6611 th3qlem2 6612 mulcmpblnq 7317 addpipqqs 7319 ordpipqqs 7323 enq0breq 7385 mulcmpblnq0 7393 nqpnq0nq 7402 nqnq0a 7403 nqnq0m 7404 nq0m0r 7405 nq0a0 7406 distrlem5prl 7535 distrlem5pru 7536 addcmpblnr 7688 ltsrprg 7696 mulgt0sr 7727 add20 8380 cru 8508 qaddcl 9581 qmulcl 9583 xaddval 9789 xnn0xadd0 9811 fzopth 10004 modqval 10267 seqvalcd 10402 seqovcd 10406 1exp 10492 m1expeven 10510 nn0opthd 10643 faclbnd 10662 faclbnd3 10664 bcn0 10676 reval 10800 absval 10952 clim 11231 fsumparts 11420 dvds2add 11774 dvds2sub 11775 opoe 11841 omoe 11842 opeo 11843 omeo 11844 gcddvds 11905 gcdcl 11908 gcdeq0 11919 gcdneg 11924 gcdaddm 11926 gcdabs 11930 gcddiv 11961 eucalgval2 11994 lcmabs 12017 rpmul 12039 divgcdcoprmex 12043 prmexpb 12092 rpexp 12094 nn0gcdsq 12141 pcqmul 12244 mul4sq 12333 plusfvalg 12604 0subm 12689 cnmpt2t 13046 cnmpt22f 13048 hmeofvalg 13056 bdmetval 13253 mul2sq 13705 |
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