Theorem ineq2 3368

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

Assertion

Ref	Expression
ineq2	⊢ (𝐴 = 𝐵 → (𝐶 ∩ 𝐴) = (𝐶 ∩ 𝐵))

Proof of Theorem ineq2

Step	Hyp	Ref	Expression
1		ineq1 3367	. 2 ⊢ (𝐴 = 𝐵 → (𝐴 ∩ 𝐶) = (𝐵 ∩ 𝐶))
2		incom 3365	. 2 ⊢ (𝐶 ∩ 𝐴) = (𝐴 ∩ 𝐶)
3		incom 3365	. 2 ⊢ (𝐶 ∩ 𝐵) = (𝐵 ∩ 𝐶)
4	1, 2, 3	3eqtr4g 2263	1 ⊢ (𝐴 = 𝐵 → (𝐶 ∩ 𝐴) = (𝐶 ∩ 𝐵))

Syntax hints:   → wi 4   = wceq 1373   ∩ cin 3165

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-v 2774  df-in 3172

This theorem is referenced by:  ineq12  3369  ineq2i  3371  ineq2d  3374  uneqin  3424  intprg  3918  fiintim  7028  uzin2  11298  inopn  14475  basis1  14519  basis2  14520  baspartn  14522  metreslem  14852  qtopbasss  14993