Theorem ineq12i 3358

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 i^i C ) = ( B i^i D )

Proof of Theorem ineq12i

Step	Hyp	Ref	Expression
1		ineq1i.1	. 2 |- A = B
2		ineq12i.2	. 2 |- C = D
3		ineq12 3355	. 2 |- ( ( A = B /\ C = D ) -> ( A i^i C ) = ( B i^i D ) )
4	1, 2, 3	mp2an 426	1 |- ( A i^i C ) = ( B i^i D )

Syntax hints:   = wceq 1364   i^i cin 3152

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-v 2762  df-in 3159

This theorem is referenced by:  undir  3409  difindir  3414  inrab  3431  inrab2  3432  inxp  4796  resindi  4957  resindir  4958  cnvin  5073  rnin  5075  inimass  5082  funtp  5307  imainlem  5335  imain  5336  offres  6187  djuinr  7122  djuin  7123  casefun  7144  exmidfodomrlemim  7261  enq0enq  7491  explecnv  11648