Theorem ineq1d 3322

Description: Equality deduction for intersection of two classes. (Contributed by NM, 10-Apr-1994.)

Hypothesis

Ref	Expression
ineq1d.1	⊢ (𝜑 → 𝐴 = 𝐵)

Assertion

Ref	Expression
ineq1d	⊢ (𝜑 → (𝐴 ∩ 𝐶) = (𝐵 ∩ 𝐶))

Proof of Theorem ineq1d

Step	Hyp	Ref	Expression
1		ineq1d.1	. 2 ⊢ (𝜑 → 𝐴 = 𝐵)
2		ineq1 3316	. 2 ⊢ (𝐴 = 𝐵 → (𝐴 ∩ 𝐶) = (𝐵 ∩ 𝐶))
3	1, 2	syl 14	1 ⊢ (𝜑 → (𝐴 ∩ 𝐶) = (𝐵 ∩ 𝐶))

Syntax hints:   → wi 4   = wceq 1343   ∩ cin 3115

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 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-v 2728  df-in 3122

This theorem is referenced by:  diftpsn3  3714  disji2  3975  ordpwsucexmid  4547  riinint  4865  fnresdisj  5298  fnimadisj  5308  ecinxp  6576  fiintim  6894  fival  6935  fzval2  9947  fvinim0ffz  10176  fsum1p  11359  fprod1p  11540  restopnb  12831  metrest  13156  qtopbasss  13171