Theorem ineq12i 3430

Description: Equality inference for intersection of two classes. (Contributed by NM, 24-Jun-2004.) (Proof shortened by Eric Schmidt, 26-Jan-2007.)

Hypotheses

Ref	Expression
ineq1i.1	⊢ 𝐴 = 𝐵
ineq12i.2	⊢ 𝐶 = 𝐷

Assertion

Ref	Expression
ineq12i	⊢ (𝐴 ∩ 𝐶) = (𝐵 ∩ 𝐷)

Proof of Theorem ineq12i

Step	Hyp	Ref	Expression
1		ineq1i.1	. 2 ⊢ 𝐴 = 𝐵
2		ineq12i.2	. 2 ⊢ 𝐶 = 𝐷
3		ineq12 3427	. 2 ⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → (𝐴 ∩ 𝐶) = (𝐵 ∩ 𝐷))
4	1, 2, 3	mp2an 430	1 ⊢ (𝐴 ∩ 𝐶) = (𝐵 ∩ 𝐷)

Syntax hints:   = wceq 1402   ∩ cin 3219

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-v 2823  df-in 3226

This theorem is referenced by:  undir  3481  difindir  3486  inrab  3505  inrab2  3506  inxp  4909  resindi  5073  resindir  5074  cnvin  5190  rnin  5192  inimass  5199  funtp  5429  imainlem  5457  imain  5458  offres  6358  djuinr  7393  djuin  7394  casefun  7415  exmidfodomrlemim  7543  enq0enq  7788  explecnv  12250