Theorem ineqan12d 3279

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 3272	. 2 |- ( ( A = B /\ C = D ) -> ( A i^i C ) = ( B i^i D ) )
4	1, 2, 3	syl2an 287	1 |- ( ( ph /\ ps ) -> ( A i^i C ) = ( B i^i D ) )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1331   i^i cin 3070

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-v 2688  df-in 3077

This theorem is referenced by:  fvun1  5487  fndmin  5527  offval  5989  ofrfval  5990  offval3  6032  iooinsup  11046