Theorem iuneq2d 4682

Description: Equality deduction for indexed union. (Contributed by Drahflow, 22-Oct-2015.)

Hypothesis

Ref	Expression
iuneq2d.2	⊢ (𝜑 → 𝐵 = 𝐶)

Assertion

Ref	Expression
iuneq2d	⊢ (𝜑 → ∪ 𝑥 ∈ 𝐴 𝐵 = ∪ 𝑥 ∈ 𝐴 𝐶)

Distinct variable groups:   𝜑,𝑥   𝑥,𝐴

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

Proof of Theorem iuneq2d

Step	Hyp	Ref	Expression
1		iuneq2d.2	. . 3 ⊢ (𝜑 → 𝐵 = 𝐶)
2	1	adantr 466	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 = 𝐶)
3	2	iuneq2dv 4677	1 ⊢ (𝜑 → ∪ 𝑥 ∈ 𝐴 𝐵 = ∪ 𝑥 ∈ 𝐴 𝐶)

Syntax hints:   → wi 4   = wceq 1631   ∈ wcel 2145  ∪ ciun 4655

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751

This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-v 3353  df-in 3730  df-ss 3737  df-iun 4657

This theorem is referenced by:  iununi  4745  oelim2  7833  ituniiun  9450  dfrtrclrec2  14005  rtrclreclem1  14006  rtrclreclem2  14007  rtrclreclem4  14009  imasval  16379  mreacs  16526  cnextval  22085  taylfval  24333  iunpreima  29721  reprdifc  31045  msubvrs  31795  trpredeq1  32056  trpredeq2  32057  neibastop2  32693  voliunnfl  33786  sstotbnd2  33905  equivtotbnd  33909  totbndbnd  33920  heiborlem3  33944  eliunov2  38497  fvmptiunrelexplb0d  38502  fvmptiunrelexplb1d  38504  comptiunov2i  38524  trclrelexplem  38529  dftrcl3  38538  trclfvcom  38541  cnvtrclfv  38542  cotrcltrcl  38543  trclimalb2  38544  trclfvdecomr  38546  dfrtrcl3  38551  dfrtrcl4  38556  isomenndlem  41261  ovnval  41272  hoicvr  41279  hoicvrrex  41287  ovnlecvr  41289  ovncvrrp  41295  ovnsubaddlem1  41301  hoidmvlelem3  41328  hoidmvle  41331  ovnhoilem1  41332  ovnovollem1  41387  smflimlem3  41498  otiunsndisjX  41818