Theorem ineq12d 3188

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 3182	. 2 |- ( ( A = B /\ C = D ) -> ( A i^i C ) = ( B i^i D ) )
4	1, 2, 3	syl2anc 403	1 |- ( ph -> ( A i^i C ) = ( B i^i D ) )

Syntax hints:   -> wi 4   = wceq 1287   i^i cin 2985

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067

This theorem depends on definitions:  df-bi 115  df-tru 1290  df-nf 1393  df-sb 1690  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-v 2616  df-in 2992

This theorem is referenced by:  csbing  3193  funprg  5020  funtpg  5021  offval  5801  ofrfval  5802  undifdc  6564  djudom  6708