Theorem ineq1d 3409

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 3403	. 2 ⊢ (𝐴 = 𝐵 → (𝐴 ∩ 𝐶) = (𝐵 ∩ 𝐶))
3	1, 2	syl 14	1 ⊢ (𝜑 → (𝐴 ∩ 𝐶) = (𝐵 ∩ 𝐶))

Syntax hints:   → wi 4   = wceq 1398   ∩ cin 3200

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 2213

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-v 2805  df-in 3207

This theorem is referenced by:  diftpsn3  3819  disji2  4085  ordpwsucexmid  4674  riinint  4999  fnresdisj  5449  fnimadisj  5460  ecinxp  6822  fiintim  7166  fival  7212  fzval2  10291  fvinim0ffz  10533  fsum1p  12042  fprod1p  12223  strressid  13217  restopnb  14975  metrest  15300  qtopbasss  15315