Theorem iineq2 2579

Description: Equality theorem for indexed intersection.

Assertion

Ref	Expression
iineq2	|- (A.x e. A B = C -> |^|_x e. A B = |^|_x e. A C)

Proof of Theorem iineq2

Step	Hyp	Ref	Expression
1		hbra1 1687	. . . 4 |- (A.x e. A B = C -> A.xA.x e. A B = C)
2		ra4 1694	. . . . . 6 |- (A.x e. A B = C -> (x e. A -> B = C))
3		eleq2 1535	. . . . . 6 |- (B = C -> (y e. B <-> y e. C))
4	2, 3	syl6 22	. . . . 5 |- (A.x e. A B = C -> (x e. A -> (y e. B <-> y e. C)))
5	4	imp 350	. . . 4 |- ((A.x e. A B = C /\ x e. A) -> (y e. B <-> y e. C))
6	1, 5	ralbida 1657	. . 3 |- (A.x e. A B = C -> (A.x e. A y e. B <-> A.x e. A y e. C))
7	6	abbidv 1577	. 2 |- (A.x e. A B = C -> {y | A.x e. A y e. B} = {y | A.x e. A y e. C})
8		df-iin 2569	. 2 |- |^|_x e. A B = {y | A.x e. A y e. B}
9		df-iin 2569	. 2 |- |^|_x e. A C = {y | A.x e. A y e. C}
10	7, 8, 9	3eqtr4g 1531	1 |- (A.x e. A B = C -> |^|_x e. A B = |^|_x e. A C)

Syntax hints:   -> wi 3   <-> wb 146   = wceq 956   e. wcel 958  {cab 1463  A.wral 1645  |^|_ciin 2567

This theorem is referenced by:  iineq2i 2581  iineq2dv 2583  iincld 7679

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-12 968  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 981  df-sb 1172  df-clab 1464  df-cleq 1469  df-clel 1472  df-ral 1649  df-iin 2569

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