Theorem iuneq1d 4948

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 4937	. 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 1537  ∪ ciun 4921

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ral 3145  df-rex 3146  df-v 3498  df-in 3945  df-ss 3954  df-iun 4923

This theorem is referenced by:  iuneq12d  4949  disjxiun  5065  kmlem11  9588  prmreclem4  16257  imasval  16786  iundisj  24151  iundisj2  24152  voliunlem1  24153  iunmbl  24156  volsup  24159  uniioombllem4  24189  iuninc  30314  iundisjf  30341  iundisj2f  30342  suppovss  30428  iundisjfi  30521  iundisj2fi  30522  iundisjcnt  30523  indval2  31275  sigaclcu3  31383  fiunelros  31435  meascnbl  31480  bnj1113  32059  bnj155  32153  bnj570  32179  bnj893  32202  cvmliftlem10  32543  mrsubvrs  32771  msubvrs  32809  voliunnfl  34938  volsupnfl  34939  heiborlem3  35093  heibor  35101  iunrelexp0  40054  iunp1  41335  iundjiunlem  42748  iundjiun  42749  meaiuninclem  42769  meaiuninc  42770  carageniuncllem1  42810  carageniuncllem2  42811  carageniuncl  42812  caratheodorylem1  42815  caratheodorylem2  42816