Theorem ineq2 3402

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 3401	. 2 ⊢ (𝐴 = 𝐵 → (𝐴 ∩ 𝐶) = (𝐵 ∩ 𝐶))
2		incom 3399	. 2 ⊢ (𝐶 ∩ 𝐴) = (𝐴 ∩ 𝐶)
3		incom 3399	. 2 ⊢ (𝐶 ∩ 𝐵) = (𝐵 ∩ 𝐶)
4	1, 2, 3	3eqtr4g 2289	1 ⊢ (𝐴 = 𝐵 → (𝐶 ∩ 𝐴) = (𝐶 ∩ 𝐵))

Syntax hints:   → wi 4   = 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:  ineq12  3403  ineq2i  3405  ineq2d  3408  uneqin  3458  intprg  3961  fiintim  7122  uzin2  11547  inopn  14726  basis1  14770  basis2  14771  baspartn  14773  metreslem  15103  qtopbasss  15244