Theorem ineq12 3427

Description: Equality theorem for intersection of two classes. (Contributed by NM, 8-May-1994.)

Assertion

Ref	Expression
ineq12	|- ( ( A = B /\ C = D ) -> ( A i^i C ) = ( B i^i D ) )

Proof of Theorem ineq12

Step	Hyp	Ref	Expression
1		ineq1 3425	. 2 |- ( A = B -> ( A i^i C ) = ( B i^i C ) )
2		ineq2 3426	. 2 |- ( C = D -> ( B i^i C ) = ( B i^i D ) )
3	1, 2	sylan9eq 2291	1 |- ( ( A = B /\ C = D ) -> ( A i^i C ) = ( B i^i D ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   = wceq 1402   i^i cin 3219

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This proof depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-v 2823  df-in 3226

This theorem is used by:  ineq12i  3430  ineq12d  3433  ineqan12d  3434  vvin  3569  fnun  5489  endisj  7122  sbthlemi8  7281  pm54.43  7536  epttop  15191  restbasg  15269  txbas  15359  bj-inex  16933