Theorem iineq2dv 4543

Description: Equality deduction for indexed intersection. (Contributed by NM, 3-Aug-2004.)

Hypothesis

Ref	Expression
iuneq2dv.1	|- ( ( ph /\ x e. A ) -> B = C )

Assertion

Ref	Expression
iineq2dv	|- ( ph -> |^|_ x e. A B = |^|_ x e. A C )

Distinct variable group:   ph, x

Allowed substitution hints:   A( x)   B( x)   C( x)

Proof of Theorem iineq2dv

Step	Hyp	Ref	Expression
1		nfv 1843	. 2 |- F/ x ph
2		iuneq2dv.1	. 2 |- ( ( ph /\ x e. A ) -> B = C )
3	1, 2	iineq2d 4541	1 |- ( ph -> |^|_ x e. A B = |^|_ x e. A C )

Syntax hints:   -> wi 4   /\ wa 384   = 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:  cntziinsn  17767  ptbasfi  21384  fclsval  21812  taylfval  24113  polfvalN  35190  dihglblem3N  36584  dihmeetlem2N  36588  iineq12dv  39289  saliincl  40545  iccvonmbllem  40892  vonicclem2  40898  smflimlem3  40981  smflimlem4  40982  smflimlem6  40984  smflimsuplem3  41028