Theorem inteqd 3836

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 3834	. 2 ⊢ (𝐴 = 𝐵 → ∩ 𝐴 = ∩ 𝐵)
3	1, 2	syl 14	1 ⊢ (𝜑 → ∩ 𝐴 = ∩ 𝐵)

Syntax hints:   → wi 4   = wceq 1348  ∩ cint 3831

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-int 3832

This theorem is referenced by:  intprg  3864  op1stbg  4464  onsucmin  4491  elreldm  4837  elxp5  5099  fniinfv  5554  1stval2  6134  2ndval2  6135  fundmen  6784  xpsnen  6799  fiintim  6906  elfi2  6949  fi0  6952  cardcl  7158  isnumi  7159  cardval3ex  7162  carden2bex  7166  clsfval  12895  clsval  12905