Theorem iineq2d 4973

Description: Equality deduction for indexed intersection. (Contributed by NM, 7-Dec-2011.)

Hypotheses

Ref	Expression
iineq2d.1	⊢ Ⅎ𝑥𝜑
iineq2d.2	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 = 𝐶)

Assertion

Ref	Expression
iineq2d	⊢ (𝜑 → ∩ 𝑥 ∈ 𝐴 𝐵 = ∩ 𝑥 ∈ 𝐴 𝐶)

Proof of Theorem iineq2d

Step	Hyp	Ref	Expression
1		iineq2d.1	. . 3 ⊢ Ⅎ𝑥𝜑
2		iineq2d.2	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 = 𝐶)
3	1, 2	ralrimia 3261	. 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝐵 = 𝐶)
4		iineq2 4970	. 2 ⊢ (∀𝑥 ∈ 𝐴 𝐵 = 𝐶 → ∩ 𝑥 ∈ 𝐴 𝐵 = ∩ 𝑥 ∈ 𝐴 𝐶)
5	3, 4	syl 17	1 ⊢ (𝜑 → ∩ 𝑥 ∈ 𝐴 𝐵 = ∩ 𝑥 ∈ 𝐴 𝐶)

Syntax hints:   → wi 4   ∧ wa 399   = wceq 1560  Ⅎwnf 1803   ∈ wcel 2142  ∀wral 3076  ∩ ciin 4950

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-12 2212  ax-ext 2734

This theorem depends on definitions:  df-bi 209  df-an 400  df-ex 1800  df-nf 1804  df-sb 2091  df-clab 2741  df-cleq 2754  df-clel 2837  df-ral 3077  df-iin 4952

This theorem is referenced by:  pmapglbx  40393  saliinclf  46900  smflimmpt  47384  smfsupmpt  47389  smfinfmpt  47393  smflimsuplem4  47397  smflimsupmpt  47403  smfliminfmpt  47406  iinfssclem1  49675  iinfssclem3  49677