Theorem iineq2dv 2595

Description: Equality deduction for indexed intersection.

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

Proof of Theorem iineq2dv

Step	Hyp	Ref	Expression
1		iuneq2dv.1	. . 3 |- ((ph /\ x e. A) -> B = C)
2	1	r19.21aiva 1721	. 2 |- (ph -> A.x e. A B = C)
3		iineq2 2591	. 2 |- (A.x e. A B = C -> |^|_x e. A B = |^|_x e. A C)
4	2, 3	syl 10	1 |- (ph -> |^|_x e. A B = |^|_x e. A C)

Syntax hints:   -> wi 3   /\ wa 223   = wceq 960   e. wcel 962  A.wral 1652  |^|_ciin 2579

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 966  ax-gen 967  ax-8 968  ax-10 970  ax-12 972  ax-17 975  ax-4 977  ax-5o 979  ax-6o 982  ax-9o 1129  ax-10o 1146  ax-16 1216  ax-11o 1224  ax-ext 1466

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 985  df-sb 1178  df-clab 1471  df-cleq 1476  df-clel 1479  df-ral 1656  df-iin 2581

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