Theorem inteqd 3956

Description: Equality deduction for class intersection. (Contributed by NM, 2-Sep-2003.)

Hypothesis

Ref	Expression
inteqd.1	⊢ (𝜑 → 𝐴 = 𝐵)

Assertion

Ref	Expression
inteqd	⊢ (𝜑 → ∩ 𝐴 = ∩ 𝐵)

Proof of Theorem inteqd

Step	Hyp	Ref	Expression
1		inteqd.1	. 2 ⊢ (𝜑 → 𝐴 = 𝐵)
2		inteq 3954	. 2 ⊢ (𝐴 = 𝐵 → ∩ 𝐴 = ∩ 𝐵)
3	1, 2	syl 14	1 ⊢ (𝜑 → ∩ 𝐴 = ∩ 𝐵)

Syntax hints:   → wi 4   = wceq 1398  ∩ cint 3951

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2216

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-int 3952

This theorem is referenced by:  intprg  3984  op1stbg  4602  onsucmin  4631  elreldm  4985  elxp5  5253  fniinfv  5737  1stval2  6351  2ndval2  6352  fundmen  7049  xpsnen  7074  fiintim  7193  elfi2  7261  fi0  7264  cardcl  7479  isnumi  7480  cardval3ex  7483  carden2bex  7488  lspfval  14585  lspval  14587  lsppropd  14629  clsfval  15015  clsval  15025