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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 5699 | . 2 ⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → (𝐴𝐹𝐶) = (𝐵𝐹𝐷)) | |
4 | 1, 2, 3 | syl2an 284 | 1 ⊢ ((𝜑 ∧ 𝜓) → (𝐴𝐹𝐶) = (𝐵𝐹𝐷)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 = wceq 1296 (class class class)co 5690 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 668 ax-5 1388 ax-7 1389 ax-gen 1390 ax-ie1 1434 ax-ie2 1435 ax-8 1447 ax-10 1448 ax-11 1449 ax-i12 1450 ax-bndl 1451 ax-4 1452 ax-17 1471 ax-i9 1475 ax-ial 1479 ax-i5r 1480 ax-ext 2077 |
This theorem depends on definitions: df-bi 116 df-3an 929 df-tru 1299 df-nf 1402 df-sb 1700 df-clab 2082 df-cleq 2088 df-clel 2091 df-nfc 2224 df-rex 2376 df-v 2635 df-un 3017 df-sn 3472 df-pr 3473 df-op 3475 df-uni 3676 df-br 3868 df-iota 5014 df-fv 5057 df-ov 5693 |
This theorem is referenced by: oveqan12rd 5710 offval 5901 offval3 5943 ecovdi 6443 ecovidi 6444 distrpig 6989 addcmpblnq 7023 addpipqqs 7026 mulpipq 7028 addcomnqg 7037 addcmpblnq0 7099 distrnq0 7115 recexprlem1ssl 7289 recexprlem1ssu 7290 1idsr 7411 addcnsrec 7476 mulcnsrec 7477 mulid1 7582 mulsub 7976 mulsub2 7977 muleqadd 8234 divmuldivap 8276 div2subap 8399 addltmul 8750 xnegdi 9434 fzsubel 9623 fzoval 9708 mulexp 10125 sqdivap 10150 crim 10423 readd 10434 remullem 10436 imadd 10442 cjadd 10449 cjreim 10468 sqrtmul 10599 sqabsadd 10619 sqabssub 10620 absmul 10633 abs2dif 10670 binom 11042 sinadd 11191 cosadd 11192 dvds2ln 11271 absmulgcd 11448 gcddiv 11450 bezoutr1 11464 lcmgcd 11502 nn0gcdsq 11620 crth 11642 rescncf 12349 |
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