ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  uneq12d Structured version   GIF version

Theorem uneq12d 3092
Description: Equality deduction for union of two classes. (Contributed by NM, 29-Sep-2004.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
Hypotheses
Ref Expression
uneq1d.1 (φA = B)
uneq12d.2 (φ𝐶 = 𝐷)
Assertion
Ref Expression
uneq12d (φ → (A𝐶) = (B𝐷))

Proof of Theorem uneq12d
StepHypRef Expression
1 uneq1d.1 . 2 (φA = B)
2 uneq12d.2 . 2 (φ𝐶 = 𝐷)
3 uneq12 3086 . 2 ((A = B 𝐶 = 𝐷) → (A𝐶) = (B𝐷))
41, 2, 3syl2anc 391 1 (φ → (A𝐶) = (B𝐷))
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1242  cun 2909
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-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
This theorem is referenced by:  disjpr2  3425  diftpsn3  3496  suceq  4105  rnpropg  4743  fntpg  4898  foun  5088  fnimapr  5176  fprg  5289  fsnunfv  5306  fsnunres  5307  tfrlemi1  5887  ereq1  6049  fztp  8670  fzsuc2  8671  fseq1p1m1  8686
  Copyright terms: Public domain W3C validator