Theorem ineq1i 3404

Description: Equality inference for intersection of two classes. (Contributed by NM, 26-Dec-1993.)

Hypothesis

Ref	Expression
ineq1i.1	⊢ 𝐴 = 𝐵

Assertion

Ref	Expression
ineq1i	⊢ (𝐴 ∩ 𝐶) = (𝐵 ∩ 𝐶)

Proof of Theorem ineq1i

Step	Hyp	Ref	Expression
1		ineq1i.1	. 2 ⊢ 𝐴 = 𝐵
2		ineq1 3401	. 2 ⊢ (𝐴 = 𝐵 → (𝐴 ∩ 𝐶) = (𝐵 ∩ 𝐶))
3	1, 2	ax-mp 5	1 ⊢ (𝐴 ∩ 𝐶) = (𝐵 ∩ 𝐶)

Syntax hints:   = wceq 1397   ∩ cin 3199

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-v 2804  df-in 3206

This theorem is referenced by:  in12  3418  inindi  3424  dfrab2  3482  dfrab3  3483  disjpr2  3733  resres  5025  imainrect  5182  ssenen  7036  minmax  11790  xrminmax  11825  nnmindc  12604  nnminle  12605  setsfun  13116  setsfun0  13117  ressressg  13157  tgrest  14892