Theorem iuneq1d 4962

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

Hypothesis

Ref	Expression
iuneq1d.1	⊢ (𝜑 → 𝐴 = 𝐵)

Assertion

Ref	Expression
iuneq1d	⊢ (𝜑 → ∪ 𝑥 ∈ 𝐴 𝐶 = ∪ 𝑥 ∈ 𝐵 𝐶)

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

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

Proof of Theorem iuneq1d

Step	Hyp	Ref	Expression
1		iuneq1d.1	. 2 ⊢ (𝜑 → 𝐴 = 𝐵)
2		iuneq1 4951	. 2 ⊢ (𝐴 = 𝐵 → ∪ 𝑥 ∈ 𝐴 𝐶 = ∪ 𝑥 ∈ 𝐵 𝐶)
3	1, 2	syl 17	1 ⊢ (𝜑 → ∪ 𝑥 ∈ 𝐴 𝐶 = ∪ 𝑥 ∈ 𝐵 𝐶)

Syntax hints:   → wi 4   = wceq 1542  ∪ ciun 4934

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1545  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-rex 3063  df-v 3432  df-ss 3907  df-iun 4936

This theorem is referenced by:  iuneq12dOLD  4963  disjxiun  5083  kmlem11  10077  indval2  12158  prmreclem4  16884  imasval  17469  iundisj  25528  iundisj2  25529  voliunlem1  25530  iunmbl  25533  volsup  25536  uniioombllem4  25566  iuninc  32648  iundisjf  32677  iundisj2f  32678  suppovss  32772  iundisjfi  32887  iundisj2fi  32888  iundisjcnt  32889  sigaclcu3  34285  fiunelros  34337  meascnbl  34382  bnj1113  34947  bnj155  35040  bnj570  35066  bnj893  35089  cvmliftlem10  35495  mrsubvrs  35723  msubvrs  35761  voliunnfl  38002  volsupnfl  38003  heiborlem3  38151  heibor  38159  iunrelexp0  44150  iunp1  45518  iundjiunlem  46908  iundjiun  46909  meaiuninclem  46929  meaiuninc  46930  carageniuncllem1  46970  carageniuncllem2  46971  carageniuncl  46972  caratheodorylem1  46975  caratheodorylem2  46976  imasubclem3  49596  imaf1hom  49598