Theorem iuneq2dv 4732

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 3147	. 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝐵 = 𝐶)
3		iuneq2 4727	. 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 385   = wceq 1653   ∈ wcel 2157  ∀wral 3089  ∪ ciun 4710

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-ext 2777

This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ral 3094  df-rex 3095  df-v 3387  df-in 3776  df-ss 3783  df-iun 4712

This theorem is referenced by:  iuneq12d  4736  iuneq2d  4737  fparlem3  7516  fparlem4  7517  oalim  7852  omlim  7853  oelim  7854  oelim2  7915  r1val3  8951  imasdsval  16490  acsfn  16634  tgidm  21113  cmpsub  21532  alexsublem  22176  bcth3  23457  ovoliunlem1  23610  voliunlem1  23658  uniiccdif  23686  uniioombllem2  23691  uniioombllem3a  23692  uniioombllem4  23694  itg2monolem1  23858  taylpfval  24460  ofpreima2  29985  esum2dlem  30670  eulerpartlemgu  30955  cvmscld  31772  msubvrs  31974  mblfinlem2  33936  ftc1anclem6  33978  heibor  34107  trclfvcom  38794  meaiininclem  41442  carageniuncllem2  41478  hoidmv1le  41550  hoidmvle  41556  ovnhoilem2  41558  ovnhoi  41559  ovnlecvr2  41566  ovncvr2  41567  hspmbl  41585  ovolval4lem1  41605  ovnovollem1  41612  ovnovollem2  41613  iunhoiioo  41632  vonioolem2  41637  smflimlem4  41724  smflimlem6  41726