Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  ineqan12d Structured version   Visualization version   GIF version

Theorem ineqan12d 4043
 Description: Equality deduction for intersection of two classes. (Contributed by NM, 7-Feb-2007.)
Hypotheses
Ref Expression
ineq1d.1 (𝜑𝐴 = 𝐵)
ineqan12d.2 (𝜓𝐶 = 𝐷)
Assertion
Ref Expression
ineqan12d ((𝜑𝜓) → (𝐴𝐶) = (𝐵𝐷))

Proof of Theorem ineqan12d
StepHypRef Expression
1 ineq1d.1 . 2 (𝜑𝐴 = 𝐵)
2 ineqan12d.2 . 2 (𝜓𝐶 = 𝐷)
3 ineq12 4036 . 2 ((𝐴 = 𝐵𝐶 = 𝐷) → (𝐴𝐶) = (𝐵𝐷))
41, 2, 3syl2an 591 1 ((𝜑𝜓) → (𝐴𝐶) = (𝐵𝐷))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 386   = wceq 1658   ∩ cin 3797 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-ext 2803 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-v 3416  df-in 3805 This theorem is referenced by:  funprg  6176  funtpg  6177  funcnvpr  6184  funcnvqp  6186  fvun1  6516  fndmin  6573  offval  7164  ofrfval  7165  offval3  7422  fpar  7545  wfrlem4  7683  wfrlem4OLD  7684  fisn  8602  ixxin  12480  vdwmc  16053  fvcosymgeq  18199  cssincl  20395  inmbl  23708  iundisj2  23715  itg1addlem3  23864  fh1  29032  iundisj2f  29950  iundisj2fi  30103  br1cosscnvxrn  34772  offval0  43146
 Copyright terms: Public domain W3C validator