MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  iuneq12d Structured version   Unicode version

Theorem iuneq12d 4109
Description: Equality deduction for indexed union, deduction version. (Contributed by Drahflow, 22-Oct-2015.)
Hypotheses
Ref Expression
iuneq1d.1  |-  ( ph  ->  A  =  B )
iuneq12d.2  |-  ( ph  ->  C  =  D )
Assertion
Ref Expression
iuneq12d  |-  ( ph  ->  U_ x  e.  A  C  =  U_ x  e.  B  D )
Distinct variable groups:    x, A    x, B    ph, x
Allowed substitution hints:    C( x)    D( x)

Proof of Theorem iuneq12d
StepHypRef Expression
1 iuneq1d.1 . . 3  |-  ( ph  ->  A  =  B )
21iuneq1d 4108 . 2  |-  ( ph  ->  U_ x  e.  A  C  =  U_ x  e.  B  C )
3 iuneq12d.2 . . . 4  |-  ( ph  ->  C  =  D )
43adantr 452 . . 3  |-  ( (
ph  /\  x  e.  B )  ->  C  =  D )
54iuneq2dv 4106 . 2  |-  ( ph  ->  U_ x  e.  B  C  =  U_ x  e.  B  D )
62, 5eqtrd 2467 1  |-  ( ph  ->  U_ x  e.  A  C  =  U_ x  e.  B  D )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1652    e. wcel 1725   U_ciun 4085
This theorem is referenced by:  cfsmolem  8142  cfsmo  8143  wunex2  8605  wuncval2  8614  imasval  13729  lpival  16308  cnextval  18084  cnextfval  18085  dvfval  19776  mblfinlem  26234  heiborlem10  26520  otiunsndisj  28056  2spotiundisj  28388
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ral 2702  df-rex 2703  df-v 2950  df-in 3319  df-ss 3326  df-iun 4087
  Copyright terms: Public domain W3C validator