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

Theorem dmeqd 4480
Description: Equality deduction for domain. (Contributed by NM, 4-Mar-2004.)
Hypothesis
Ref Expression
dmeqd.1 (φA = B)
Assertion
Ref Expression
dmeqd (φ → dom A = dom B)

Proof of Theorem dmeqd
StepHypRef Expression
1 dmeqd.1 . 2 (φA = B)
2 dmeq 4478 . 2 (A = B → dom A = dom B)
31, 2syl 14 1 (φ → dom A = dom B)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1242  dom cdm 4288
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-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-sn 3373  df-pr 3374  df-op 3376  df-br 3756  df-dm 4298
This theorem is referenced by:  rneq  4504  dmsnsnsng  4741  elxp4  4751  fndmin  5217  1stvalg  5711  fo1st  5726  f1stres  5728  errn  6064  xpassen  6240  xpdom2  6241
  Copyright terms: Public domain W3C validator