Theorem ineq12i 4159

Description: Equality inference for intersection of two classes. (Contributed by NM, 24-Jun-2004.) (Proof shortened by Eric Schmidt, 26-Jan-2007.)

Hypotheses

Ref	Expression
ineq1i.1	⊢ 𝐴 = 𝐵
ineq12i.2	⊢ 𝐶 = 𝐷

Assertion

Ref	Expression
ineq12i	⊢ (𝐴 ∩ 𝐶) = (𝐵 ∩ 𝐷)

Proof of Theorem ineq12i

Step	Hyp	Ref	Expression
1		ineq1i.1	. 2 ⊢ 𝐴 = 𝐵
2		ineq12i.2	. 2 ⊢ 𝐶 = 𝐷
3		ineq12 4156	. 2 ⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → (𝐴 ∩ 𝐶) = (𝐵 ∩ 𝐷))
4	1, 2, 3	mp2an 693	1 ⊢ (𝐴 ∩ 𝐶) = (𝐵 ∩ 𝐷)

Syntax hints:   = wceq 1542   ∩ cin 3889

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1545  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-rab 3391  df-in 3897

This theorem is referenced by:  undir  4228  difundi  4231  difindir  4234  inrab  4257  inrab2  4258  elneldisj  4333  dfif4  4483  dfif5  4484  inxpOLD  5782  resindi  5955  resindir  5956  rnin  6105  inimass  6114  cnvrescnv  6154  predin  6286  funtp  6550  orduniss2  7778  offres  7930  fodomr  9060  fodomfir  9232  epinid0  9513  cnvepnep  9523  wemapwe  9612  cotr3  14934  explecnv  15824  psssdm2  18541  ablfacrp  20037  cnfldfunALT  21362  cnfldfunALTOLD  21375  pjfval2  21702  ofco2  22429  iundisj2  25529  clwwlknondisj  30199  lejdiri  31628  cmbr3i  31689  nonbooli  31740  5oai  31750  3oalem5  31755  mayetes3i  31818  mdexchi  32424  disjpreima  32672  disjxpin  32676  iundisj2f  32678  xppreima  32736  iundisj2fi  32888  xpinpreima  34069  xpinpreima2  34070  ordtcnvNEW  34083  pprodcnveq  36082  dfiota3  36122  bj-inrab  37253  ptrest  37957  ftc1anclem6  38036  dmxrn  38725  xrnres3  38765  br2coss  38866  1cosscnvxrn  38903  refsymrels2  38987  dfeqvrels2  39010  dfeldisj5  39151  dnwech  43497  fgraphopab  43652  onfrALTlem5  44990  onfrALTlem4  44991  onfrALTlem5VD  45332  onfrALTlem4VD  45333  disjxp1  45521  disjinfi  45643  oppczeroo  49727