Theorem ineq12i 3408

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 3405	. 2 ⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → (𝐴 ∩ 𝐶) = (𝐵 ∩ 𝐷))
4	1, 2, 3	mp2an 426	1 ⊢ (𝐴 ∩ 𝐶) = (𝐵 ∩ 𝐷)

Syntax hints:   = wceq 1398   ∩ cin 3200

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-v 2805  df-in 3207

This theorem is referenced by:  undir  3459  difindir  3464  inrab  3481  inrab2  3482  inxp  4870  resindi  5034  resindir  5035  cnvin  5151  rnin  5153  inimass  5160  funtp  5390  imainlem  5418  imain  5419  offres  6306  djuinr  7305  djuin  7306  casefun  7327  exmidfodomrlemim  7455  enq0enq  7694  explecnv  12129