Theorem iuneq2dv 5020

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

Hypothesis

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

Assertion

Ref	Expression
iuneq2dv	⊢ (𝜑 → ∪ 𝑥 ∈ 𝐴 𝐵 = ∪ 𝑥 ∈ 𝐴 𝐶)

Distinct variable group:   𝜑,𝑥

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

Proof of Theorem iuneq2dv

Step	Hyp	Ref	Expression
1		iuneq2dv.1	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 = 𝐶)
2	1	ralrimiva 3144	. 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝐵 = 𝐶)
3		iuneq2 5015	. 2 ⊢ (∀𝑥 ∈ 𝐴 𝐵 = 𝐶 → ∪ 𝑥 ∈ 𝐴 𝐵 = ∪ 𝑥 ∈ 𝐴 𝐶)
4	2, 3	syl 17	1 ⊢ (𝜑 → ∪ 𝑥 ∈ 𝐴 𝐵 = ∪ 𝑥 ∈ 𝐴 𝐶)

Syntax hints:   → wi 4   ∧ wa 394   = wceq 1539   ∈ wcel 2104  ∀wral 3059  ∪ ciun 4996

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-ext 2701

This theorem depends on definitions:  df-bi 206  df-an 395  df-tru 1542  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2722  df-clel 2808  df-ral 3060  df-rex 3069  df-v 3474  df-in 3954  df-ss 3964  df-iun 4998

This theorem is referenced by:  iuneq12d  5024  iuneq2d  5025  fparlem3  8102  fparlem4  8103  oalim  8534  omlim  8535  oelim  8536  oelim2  8597  r1val3  9835  imasdsval  17465  acsfn  17607  tgidm  22703  cmpsub  23124  alexsublem  23768  bcth3  25079  ovoliunlem1  25251  voliunlem1  25299  uniiccdif  25327  uniioombllem2  25332  uniioombllem3a  25333  uniioombllem4  25335  itg2monolem1  25500  taylpfval  26113  ofpreima2  32158  fnpreimac  32163  esum2dlem  33388  eulerpartlemgu  33674  cvmscld  34562  satom  34645  msubvrs  34849  mblfinlem2  36829  ftc1anclem6  36869  heibor  36992  prjspval2  41657  trclfvcom  42776  scottrankd  43309  meaiininclem  45500  carageniuncllem2  45536  hoidmv1le  45608  hoidmvle  45614  ovnhoilem2  45616  ovnhoi  45617  ovnlecvr2  45624  ovncvr2  45625  hspmbl  45643  ovolval4lem1  45663  ovnovollem1  45670  ovnovollem2  45671  iunhoiioo  45690  vonioolem2  45695  smflimlem4  45788  smflimlem6  45790