Theorem iineq2dv 4936

Description: Equality deduction for indexed intersection. (Contributed by NM, 3-Aug-2004.)

Hypothesis

Ref	Expression
iuneq2dv.1	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 = 𝐶)

Assertion

Ref	Expression
iineq2dv	⊢ (𝜑 → ∩ 𝑥 ∈ 𝐴 𝐵 = ∩ 𝑥 ∈ 𝐴 𝐶)

Distinct variable group:   𝜑,𝑥

Allowed substitution hints:   𝐴(𝑥)   𝐵(𝑥)   𝐶(𝑥)

Proof of Theorem iineq2dv

Step	Hyp	Ref	Expression
1		nfv 1911	. 2 ⊢ Ⅎ𝑥𝜑
2		iuneq2dv.1	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 = 𝐶)
3	1, 2	iineq2d 4934	1 ⊢ (𝜑 → ∩ 𝑥 ∈ 𝐴 𝐵 = ∩ 𝑥 ∈ 𝐴 𝐶)

Syntax hints:   → wi 4   ∧ wa 398   = wceq 1533   ∈ wcel 2110  ∩ ciin 4912

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 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-12 2173  ax-ext 2793

This theorem depends on definitions:  df-bi 209  df-an 399  df-ex 1777  df-nf 1781  df-sb 2066  df-clab 2800  df-cleq 2814  df-clel 2893  df-ral 3143  df-iin 4914

This theorem is referenced by:  cntziinsn  18459  ptbasfi  22183  fclsval  22610  taylfval  24941  polfvalN  37034  dihglblem3N  38425  dihmeetlem2N  38429  iineq12dv  41365  saliincl  42604  iccvonmbllem  42954  vonicclem2  42960  smflimlem3  43043  smflimlem4  43044  smflimlem6  43046  smflimsuplem3  43090