Theorem ineq2 3354

Description: Equality theorem for intersection of two classes. (Contributed by NM, 26-Dec-1993.)

Assertion

Ref	Expression
ineq2	|- ( A = B -> ( C i^i A ) = ( C i^i B ) )

Proof of Theorem ineq2

Step	Hyp	Ref	Expression
1		ineq1 3353	. 2 |- ( A = B -> ( A i^i C ) = ( B i^i C ) )
2		incom 3351	. 2 |- ( C i^i A ) = ( A i^i C )
3		incom 3351	. 2 |- ( C i^i B ) = ( B i^i C )
4	1, 2, 3	3eqtr4g 2251	1 |- ( A = B -> ( C i^i A ) = ( C i^i B ) )

Syntax hints:   -> wi 4   = wceq 1364   i^i cin 3152

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 2175

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-v 2762  df-in 3159

This theorem is referenced by:  ineq12  3355  ineq2i  3357  ineq2d  3360  uneqin  3410  intprg  3903  fiintim  6985  uzin2  11131  inopn  14171  basis1  14215  basis2  14216  baspartn  14218  metreslem  14548  qtopbasss  14689