Theorem iineq2dv 2580

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 1713	. 2 |- (ph -> A.x e. A B = C)
3		iineq2 2576	. 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 955   e. wcel 957  A.wral 1644  |^|_ciin 2564

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 961  ax-gen 962  ax-8 963  ax-10 965  ax-12 967  ax-17 970  ax-4 972  ax-5o 974  ax-6o 977  ax-9o 1122  ax-10o 1139  ax-16 1210  ax-11o 1218  ax-ext 1459

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 980  df-sb 1172  df-clab 1464  df-cleq 1469  df-clel 1472  df-ral 1648  df-iin 2566

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