Theorem iineq2i 3946

Description: Equality inference for indexed intersection. (Contributed by NM, 22-Oct-2003.)

Hypothesis

Ref	Expression
iuneq2i.1	|- ( x e. A -> B = C )

Assertion

Ref	Expression
iineq2i	|- |^|_ x e. A B = |^|_ x e. A C

Proof of Theorem iineq2i

Step	Hyp	Ref	Expression
1		iineq2 3944	. 2 |- ( A. x e. A B = C -> |^|_ x e. A B = |^|_ x e. A C )
2		iuneq2i.1	. 2 |- ( x e. A -> B = C )
3	1, 2	mprg 2563	1 |- |^|_ x e. A B = |^|_ x e. A C

Syntax hints:   -> wi 4   = wceq 1373   e. wcel 2176   |^|_ciin 3928

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-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-11 1529  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-ral 2489  df-iin 3930

This theorem is referenced by:  iinrabm  3990  iinin1m  3997