Theorem ineq12d 3352

Description: Equality deduction for intersection of two classes. (Contributed by NM, 24-Jun-2004.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)

Hypotheses

Ref	Expression
ineq1d.1	|- ( ph -> A = B )
ineq12d.2	|- ( ph -> C = D )

Assertion

Ref	Expression
ineq12d	|- ( ph -> ( A i^i C ) = ( B i^i D ) )

Proof of Theorem ineq12d

Step	Hyp	Ref	Expression
1		ineq1d.1	. 2 |- ( ph -> A = B )
2		ineq12d.2	. 2 |- ( ph -> C = D )
3		ineq12 3346	. 2 |- ( ( A = B /\ C = D ) -> ( A i^i C ) = ( B i^i D ) )
4	1, 2, 3	syl2anc 411	1 |- ( ph -> ( A i^i C ) = ( B i^i D ) )

Syntax hints:   -> wi 4   = wceq 1364   i^i cin 3143

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 2171

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-v 2754  df-in 3150

This theorem is referenced by:  csbing  3357  funprg  5288  funtpg  5289  offval  6118  ofrfval  6119  undifdc  6956  djudom  7126  isunitd  13481  dfrhm2  13529  isrhm  13533  rhmval  13548  2idlvalg  13842