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Mirrors > Home > ILE Home > Th. List > oveqan12d | GIF version |
Description: Equality deduction for operation value. (Contributed by NM, 10-Aug-1995.) |
Ref | Expression |
---|---|
oveq1d.1 | ⊢ (𝜑 → 𝐴 = 𝐵) |
opreqan12i.2 | ⊢ (𝜓 → 𝐶 = 𝐷) |
Ref | Expression |
---|---|
oveqan12d | ⊢ ((𝜑 ∧ 𝜓) → (𝐴𝐹𝐶) = (𝐵𝐹𝐷)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | oveq1d.1 | . 2 ⊢ (𝜑 → 𝐴 = 𝐵) | |
2 | opreqan12i.2 | . 2 ⊢ (𝜓 → 𝐶 = 𝐷) | |
3 | oveq12 5905 | . 2 ⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → (𝐴𝐹𝐶) = (𝐵𝐹𝐷)) | |
4 | 1, 2, 3 | syl2an 289 | 1 ⊢ ((𝜑 ∧ 𝜓) → (𝐴𝐹𝐶) = (𝐵𝐹𝐷)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 = wceq 1364 (class class class)co 5896 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-ext 2171 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-nf 1472 df-sb 1774 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-rex 2474 df-v 2754 df-un 3148 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-br 4019 df-iota 5196 df-fv 5243 df-ov 5899 |
This theorem is referenced by: oveqan12rd 5916 offval 6114 offval3 6159 ecovdi 6672 ecovidi 6673 distrpig 7362 addcmpblnq 7396 addpipqqs 7399 mulpipq 7401 addcomnqg 7410 addcmpblnq0 7472 distrnq0 7488 recexprlem1ssl 7662 recexprlem1ssu 7663 1idsr 7797 addcnsrec 7871 mulcnsrec 7872 mulrid 7984 mulsub 8388 mulsub2 8389 muleqadd 8655 divmuldivap 8699 div2subap 8824 addltmul 9185 xnegdi 9898 fzsubel 10090 fzoval 10178 mulexp 10590 sqdivap 10615 crim 10899 readd 10910 remullem 10912 imadd 10918 cjadd 10925 cjreim 10944 sqrtmul 11076 sqabsadd 11096 sqabssub 11097 absmul 11110 abs2dif 11147 binom 11524 sinadd 11776 cosadd 11777 dvds2ln 11863 absmulgcd 12050 gcddiv 12052 bezoutr1 12066 lcmgcd 12110 nn0gcdsq 12232 crth 12256 pythagtriplem1 12297 pcqmul 12335 4sqlem4a 12423 4sqlem4 12424 idmhm 12921 resmhm 12939 eqgval 13162 idghm 13198 resghm 13199 isrhm 13508 rhmval 13523 xmetxp 14467 xmetxpbl 14468 txmetcnp 14478 divcnap 14515 rescncf 14528 relogoprlem 14749 lgsdir2 14895 |
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