Theorem ineq12i 4226

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 4223	. 2 ⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → (𝐴 ∩ 𝐶) = (𝐵 ∩ 𝐷))
4	1, 2, 3	mp2an 692	1 ⊢ (𝐴 ∩ 𝐶) = (𝐵 ∩ 𝐷)

Syntax hints:   = wceq 1537   ∩ cin 3962

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1540  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-rab 3434  df-in 3970

This theorem is referenced by:  undir  4293  difundi  4296  difindir  4299  inrab  4322  inrab2  4323  elneldisj  4398  dfif4  4546  dfif5  4547  inxpOLD  5846  resindi  6016  resindir  6017  rnin  6169  inimass  6177  cnvrescnv  6217  predin  6350  funtp  6625  orduniss2  7853  offres  8007  fodomr  9167  fodomfir  9366  epinid0  9638  cnvepnep  9646  wemapwe  9735  cotr3  15014  explecnv  15898  psssdm2  18639  ablfacrp  20101  cnfldfunALT  21397  cnfldfunALTOLD  21410  cnfldfunALTOLDOLD  21411  pjfval2  21747  ofco2  22473  iundisj2  25598  clwwlknondisj  30140  lejdiri  31568  cmbr3i  31629  nonbooli  31680  5oai  31690  3oalem5  31695  mayetes3i  31758  mdexchi  32364  disjpreima  32604  disjxpin  32608  iundisj2f  32610  xppreima  32662  iundisj2fi  32805  xpinpreima  33867  xpinpreima2  33868  ordtcnvNEW  33881  pprodcnveq  35865  dfiota3  35905  bj-inrab  36910  ptrest  37606  ftc1anclem6  37685  xrnres3  38386  br2coss  38420  1cosscnvxrn  38457  refsymrels2  38547  dfeqvrels2  38570  dfeldisj5  38703  dnwech  43037  fgraphopab  43192  onfrALTlem5  44540  onfrALTlem4  44541  onfrALTlem5VD  44883  onfrALTlem4VD  44884  disjxp1  45009  disjinfi  45135