Theorem iineq2dv 4935

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 1909	. 2 ⊢ Ⅎ𝑥𝜑
2		iuneq2dv.1	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 = 𝐶)
3	1, 2	iineq2d 4933	1 ⊢ (𝜑 → ∩ 𝑥 ∈ 𝐴 𝐵 = ∩ 𝑥 ∈ 𝐴 𝐶)

Syntax hints:   → wi 4   ∧ wa 398   = wceq 1531   ∈ wcel 2108  ∩ ciin 4911

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1905  ax-6 1964  ax-7 2009  ax-8 2110  ax-9 2118  ax-12 2170  ax-ext 2791

This theorem depends on definitions:  df-bi 209  df-an 399  df-ex 1775  df-nf 1779  df-sb 2064  df-clab 2798  df-cleq 2812  df-clel 2891  df-ral 3141  df-iin 4913

This theorem is referenced by:  cntziinsn  18457  ptbasfi  22181  fclsval  22608  taylfval  24939  polfvalN  37032  dihglblem3N  38423  dihmeetlem2N  38427  iineq12dv  41363  saliincl  42601  iccvonmbllem  42951  vonicclem2  42957  smflimlem3  43040  smflimlem4  43041  smflimlem6  43043  smflimsuplem3  43087