Theorem ineqan12d 3816

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 3809	. 2 |- ( ( A = B /\ C = D ) -> ( A i^i C ) = ( B i^i D ) )
4	1, 2, 3	syl2an 494	1 |- ( ( ph /\ ps ) -> ( A i^i C ) = ( B i^i D ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   i^i cin 3573

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-v 3202  df-in 3581

This theorem is referenced by:  funprg  5940  funtpg  5942  funcnvpr  5950  funcnvqp  5952  funcnvqpOLD  5953  fvun1  6269  fndmin  6324  offval  6904  ofrfval  6905  offval3  7162  fpar  7281  wfrlem4  7418  fisn  8333  ixxin  12192  vdwmc  15682  fvcosymgeq  17849  cssincl  20032  inmbl  23310  iundisj2  23317  itg1addlem3  23465  fh1  28477  iundisj2f  29403  iundisj2fi  29556  offval0  42299