Theorem iineq2 2007

Description: Equality theorem for indexed intersection.

Assertion

Ref	Expression
iineq2	⊢ (∀x ∈ A B = C → ∩x ∈ A B = ∩x ∈ A C)

Proof of Theorem iineq2

Step	Hyp	Ref	Expression
1		hbra1 1237	. . . 4 ⊢ (∀x ∈ A B = C → ∀x∀x ∈ A B = C)
2		ra4 1243	. . . . . 6 ⊢ (∀x ∈ A B = C → (x ∈ A → B = C))
3		eleq2 1150	. . . . . 6 ⊢ (B = C → (y ∈ B ↔ y ∈ C))
4	2, 3	syl6 23	. . . . 5 ⊢ (∀x ∈ A B = C → (x ∈ A → (y ∈ B ↔ y ∈ C)))
5	4	imp 277	. . . 4 ⊢ ((∀x ∈ A B = C ∧ x ∈ A) → (y ∈ B ↔ y ∈ C))
6	1, 5	biralda 1213	. . 3 ⊢ (∀x ∈ A B = C → (∀x ∈ A y ∈ B ↔ ∀x ∈ A y ∈ C))
7	6	biabdv 1183	. 2 ⊢ (∀x ∈ A B = C → {y∣∀x ∈ A y ∈ B} = {y∣∀x ∈ A y ∈ C})
8		df-iin 1997	. 2 ⊢ ∩x ∈ A B = {y∣∀x ∈ A y ∈ B}
9		df-iin 1997	. 2 ⊢ ∩x ∈ A C = {y∣∀x ∈ A y ∈ C}
10	7, 8, 9	3eqtr4g 1147	1 ⊢ (∀x ∈ A B = C → ∩x ∈ A B = ∩x ∈ A C)

Syntax hints:   → wi 2   ↔ wb 127  {cab 1090   = wceq 1091   ∈ wcel 1092  ∀wral 1201  ∩ciin 1995

This theorem is referenced by:  iineq2i 2009  iineq2dv 2011

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-ral 1205  df-iin 1997

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