Theorem iineq2dv 3934

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 1539	. 2 |- F/ x ph
2		iuneq2dv.1	. 2 |- ( ( ph /\ x e. A ) -> B = C )
3	1, 2	iineq2d 3932	1 |- ( ph -> |^|_ x e. A B = |^|_ x e. A C )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1364   e. wcel 2164   |^|_ciin 3913

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-11 1517  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-ral 2477  df-iin 3915

This theorem is referenced by: (None)