Theorem iineq2i 2009

Description: Equality inference for indexed intersection.

Hypothesis

Ref	Expression
iuneq2i.1	⊢ (x ∈ A → B = C)

Assertion

Ref	Expression
iineq2i	⊢ ∩x ∈ A B = ∩x ∈ A C

Proof of Theorem iineq2i

Step	Hyp	Ref	Expression
1		iineq2 2007	. 2 ⊢ (∀x ∈ A B = C → ∩x ∈ A B = ∩x ∈ A C)
2		iuneq2i.1	. 2 ⊢ (x ∈ A → B = C)
3	1, 2	mprg 1249	1 ⊢ ∩x ∈ A B = ∩x ∈ A C

Syntax hints:   → wi 2   = wceq 1091   ∈ wcel 1092  ∩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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