Theorem iineq2dv 3700

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 1461	. 2 |- F/ x ph
2		iuneq2dv.1	. 2 |- ( ( ph /\ x e. A ) -> B = C )
3	1, 2	iineq2d 3698	1 |- ( ph -> |^|_ x e. A B = |^|_ x e. A C )

Syntax hints:   -> wi 4   /\ wa 102   = wceq 1284   e. wcel 1433   |^|_ciin 3679

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-11 1437  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-ral 2353  df-iin 3681

This theorem is referenced by: (None)