Theorem iineq2dv 3871

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 1508	. 2 |- F/ x ph
2		iuneq2dv.1	. 2 |- ( ( ph /\ x e. A ) -> B = C )
3	1, 2	iineq2d 3869	1 |- ( ph -> |^|_ x e. A B = |^|_ x e. A C )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1335   e. wcel 2128   |^|_ciin 3850

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-11 1486  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139

This theorem depends on definitions:  df-bi 116  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-ral 2440  df-iin 3852

This theorem is referenced by: (None)