Theorem iineq2i 3989

Description: Equality inference for indexed intersection. (Contributed by NM, 22-Oct-2003.)

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 3987	. 2 ⊢ (∀x ∈ A B = C → ∩x ∈ A B = ∩x ∈ A C)
2		iuneq2i.1	. 2 ⊢ (x ∈ A → B = C)
3	1, 2	mprg 2684	1 ⊢ ∩x ∈ A B = ∩x ∈ A C

Syntax hints:   → wi 4   = wceq 1642   ∈ wcel 1710  ∩ciin 3971

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-ral 2620  df-iin 3973

This theorem is referenced by:  iinrab  4029  iinin1  4038