Theorem ineq12d 3383

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 3377	. 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 1373   i^i cin 3173

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-v 2778  df-in 3180

This theorem is referenced by:  csbing  3388  funprg  5343  funtpg  5344  offval  6189  ofrfval  6190  undifdc  7047  djudom  7221  isunitd  13983  dfrhm2  14031  isrhm  14035  rhmval  14050  2idlval  14379  2idlvalg  14380