Theorem iuneq2dv 4976

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 3154	. 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝐵 = 𝐶)
3		iuneq2 4971	. 2 ⊢ (∀𝑥 ∈ 𝐴 𝐵 = 𝐶 → ∪ 𝑥 ∈ 𝐴 𝐵 = ∪ 𝑥 ∈ 𝐴 𝐶)
4	2, 3	syl 18	1 ⊢ (𝜑 → ∪ 𝑥 ∈ 𝐴 𝐵 = ∪ 𝑥 ∈ 𝐴 𝐶)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2145  ∀wral 3076  ∪ ciun 4951

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2739  df-cleq 2752  df-clel 2835  df-ral 3077  df-rex 3087  df-v 3452  df-ss 3916  df-iun 4953

This theorem is used by:  iuneq12d  4980  iuneq2d  4981  fparlem3  8112  fparlem4  8113  oalim  8520  omlim  8521  oelim  8522  oelim2  8584  r1val3  9821  scottrankd  9889  imasdsval  17602  acsfn  17748  ssdifidllem  21548  tgidm  23206  cmpsub  23626  alexsublem  24271  bcth3  25560  ovoliunlem1  25731  voliunlem1  25779  uniiccdif  25807  uniioombllem2  25812  uniioombllem3a  25813  uniioombllem4  25815  itg2monolem1  25979  taylpfval  26602  dmdju  33121  ofpreima2  33140  fnpreimac  33144  esum2dlem  34603  eulerpartlemgu  34889  cvmscld  35853  satom  35936  msubvrs  36140  mblfinlem2  38408  ftc1anclem6  38448  heibor  38572  prjspval2  43460  trclfvcom  44564  meaiininclem  47315  carageniuncllem2  47351  hoidmv1le  47423  hoidmvle  47429  ovnhoilem2  47431  ovnhoi  47432  ovnlecvr2  47439  ovncvr2  47440  hspmbl  47458  ovolval4lem1  47478  ovnovollem1  47485  ovnovollem2  47486  iunhoiioo  47505  vonioolem2  47510  smflimlem4  47603  smflimlem6  47605