Theorem iineq1 2004

Description: Equality theorem for restricted existential quantifier.

Assertion

Ref	Expression
iineq1	⊢ (A = B → ∩x ∈ A C = ∩x ∈ B C)

Distinct variable group(s):   x,A   x,B

Proof of Theorem iineq1

Step	Hyp	Ref	Expression
1		raleq 1324	. . 3 ⊢ (A = B → (∀x ∈ A y ∈ C ↔ ∀x ∈ B y ∈ C))
2	1	biabdv 1183	. 2 ⊢ (A = B → {y∣∀x ∈ A y ∈ C} = {y∣∀x ∈ B y ∈ C})
3		df-iin 1997	. 2 ⊢ ∩x ∈ A C = {y∣∀x ∈ A y ∈ C}
4		df-iin 1997	. 2 ⊢ ∩x ∈ B C = {y∣∀x ∈ B y ∈ C}
5	2, 3, 4	3eqtr4g 1147	1 ⊢ (A = B → ∩x ∈ A C = ∩x ∈ B C)

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

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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