Theorem inteqd 3845

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 3843	. 2 ⊢ (𝐴 = 𝐵 → ∩ 𝐴 = ∩ 𝐵)
3	1, 2	syl 14	1 ⊢ (𝜑 → ∩ 𝐴 = ∩ 𝐵)

Syntax hints:   → wi 4   = wceq 1353  ∩ cint 3840

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 709  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-ext 2157

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1459  df-sb 1761  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-ral 2458  df-int 3841

This theorem is referenced by:  intprg  3873  op1stbg  4473  onsucmin  4500  elreldm  4846  elxp5  5109  fniinfv  5566  1stval2  6146  2ndval2  6147  fundmen  6796  xpsnen  6811  fiintim  6918  elfi2  6961  fi0  6964  cardcl  7170  isnumi  7171  cardval3ex  7174  carden2bex  7178  clsfval  13172  clsval  13182