Theorem iineq2dv 3895

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 1521	. 2 |- F/ x ph
2		iuneq2dv.1	. 2 |- ( ( ph /\ x e. A ) -> B = C )
3	1, 2	iineq2d 3893	1 |- ( ph -> |^|_ x e. A B = |^|_ x e. A C )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1348   e. wcel 2141   |^|_ciin 3874

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 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-11 1499  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-ral 2453  df-iin 3876

This theorem is referenced by: (None)