ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  oveqan12d Structured version   GIF version

Theorem oveqan12d 5474
Description: Equality deduction for operation value. (Contributed by NM, 10-Aug-1995.)
Hypotheses
Ref Expression
oveq1d.1 (φA = B)
opreqan12i.2 (ψ𝐶 = 𝐷)
Assertion
Ref Expression
oveqan12d ((φ ψ) → (A𝐹𝐶) = (B𝐹𝐷))

Proof of Theorem oveqan12d
StepHypRef Expression
1 oveq1d.1 . 2 (φA = B)
2 opreqan12i.2 . 2 (ψ𝐶 = 𝐷)
3 oveq12 5464 . 2 ((A = B 𝐶 = 𝐷) → (A𝐹𝐶) = (B𝐹𝐷))
41, 2, 3syl2an 273 1 ((φ ψ) → (A𝐹𝐶) = (B𝐹𝐷))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97   = wceq 1242  (class class class)co 5455
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-rex 2306  df-v 2553  df-un 2916  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-br 3756  df-iota 4810  df-fv 4853  df-ov 5458
This theorem is referenced by:  oveqan12rd  5475  offval  5661  offval3  5703  ecovdi  6153  ecovidi  6154  distrpig  6317  addcmpblnq  6351  addpipqqs  6354  mulpipq  6356  addcomnqg  6365  addcmpblnq0  6425  distrnq0  6441  recexprlem1ssl  6604  recexprlem1ssu  6605  1idsr  6676  addcnsrec  6719  mulcnsrec  6720  mulid1  6802  mulsub  7174  mulsub2  7175  muleqadd  7411  divmuldivap  7450  addltmul  7918  fzsubel  8673  fzoval  8755  mulexp  8928  sqdivap  8952  crim  9066  readd  9077  remullem  9079  imadd  9085  cjadd  9092  cjreim  9111
  Copyright terms: Public domain W3C validator