Theorem iineq2i 3989

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 3987	. 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 2589	1 |- |^|_ x e. A B = |^|_ x e. A C

Syntax hints:   -> wi 4   = wceq 1397   e. wcel 2202   |^|_ciin 3971

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 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-11 1554  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-ral 2515  df-iin 3973

This theorem is referenced by:  iinrabm  4033  iinin1m  4040