Theorem iineq2i 5019

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 5017	. 2 ⊢ (∀𝑥 ∈ 𝐴 𝐵 = 𝐶 → ∩ 𝑥 ∈ 𝐴 𝐵 = ∩ 𝑥 ∈ 𝐴 𝐶)
2		iuneq2i.1	. 2 ⊢ (𝑥 ∈ 𝐴 → 𝐵 = 𝐶)
3	1, 2	mprg 3065	1 ⊢ ∩ 𝑥 ∈ 𝐴 𝐵 = ∩ 𝑥 ∈ 𝐴 𝐶

Syntax hints:   → wi 4   = wceq 1537   ∈ wcel 2106  ∩ ciin 4997

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-iin 4999

This theorem is referenced by:  iinrab  5074  iinin1  5084  diaintclN  41041  dibintclN  41150  dihintcl  41327  imaiinfv  42681  smflimlem3  46729