Theorem iineq2i 4540

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 4538	. 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 2926	1 |- |^|_ x e. A B = |^|_ x e. A C

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   |^|_ciin 4521

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-ral 2917  df-iin 4523

This theorem is referenced by:  iinrab  4582  iinin1  4591  diaintclN  36347  dibintclN  36456  dihintcl  36633  imaiinfv  37256  smflimlem3  40981