Theorem ineqan12d 3434

Description: Equality deduction for intersection of two classes. (Contributed by NM, 7-Feb-2007.)

Hypotheses

Ref	Expression
ineq1d.1	|- ( ph -> A = B )
ineqan12d.2	|- ( ps -> C = D )

Assertion

Ref	Expression
ineqan12d	|- ( ( ph /\ ps ) -> ( A i^i C ) = ( B i^i D ) )

Proof of Theorem ineqan12d

Step	Hyp	Ref	Expression
1		ineq1d.1	. 2 |- ( ph -> A = B )
2		ineqan12d.2	. 2 |- ( ps -> C = D )
3		ineq12 3427	. 2 |- ( ( A = B /\ C = D ) -> ( A i^i C ) = ( B i^i D ) )
4	1, 2, 3	syl2an 289	1 |- ( ( ph /\ ps ) -> ( A i^i C ) = ( B i^i D ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1402   i^i 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:  fvun1  5763  fndmin  5807  offval  6300  ofrfval  6301  offval3  6357  iooinsup  12021