Theorem ineq2 3376

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 3375	. 2 ⊢ (𝐴 = 𝐵 → (𝐴 ∩ 𝐶) = (𝐵 ∩ 𝐶))
2		incom 3373	. 2 ⊢ (𝐶 ∩ 𝐴) = (𝐴 ∩ 𝐶)
3		incom 3373	. 2 ⊢ (𝐶 ∩ 𝐵) = (𝐵 ∩ 𝐶)
4	1, 2, 3	3eqtr4g 2265	1 ⊢ (𝐴 = 𝐵 → (𝐶 ∩ 𝐴) = (𝐶 ∩ 𝐵))

Syntax hints:   → wi 4   = wceq 1373   ∩ cin 3173

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 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-v 2778  df-in 3180

This theorem is referenced by:  ineq12  3377  ineq2i  3379  ineq2d  3382  uneqin  3432  intprg  3932  fiintim  7054  uzin2  11413  inopn  14590  basis1  14634  basis2  14635  baspartn  14637  metreslem  14967  qtopbasss  15108