Theorem iuneq1d 4815

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 4804	. 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 1508  ∪ ciun 4789

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1759  ax-4 1773  ax-5 1870  ax-6 1929  ax-7 1966  ax-8 2053  ax-9 2060  ax-10 2080  ax-11 2094  ax-12 2107  ax-ext 2745

This theorem depends on definitions:  df-bi 199  df-an 388  df-or 835  df-tru 1511  df-ex 1744  df-nf 1748  df-sb 2017  df-clab 2754  df-cleq 2766  df-clel 2841  df-nfc 2913  df-ral 3088  df-rex 3089  df-v 3412  df-in 3831  df-ss 3838  df-iun 4791

This theorem is referenced by:  iuneq12d  4816  disjxiun  4923  kmlem11  9379  prmreclem4  16110  imasval  16639  iundisj  23868  iundisj2  23869  voliunlem1  23870  iunmbl  23873  volsup  23876  uniioombllem4  23906  iuninc  30099  iundisjf  30123  iundisj2f  30124  suppovss  30205  iundisjfi  30293  iundisj2fi  30294  iundisjcnt  30295  indval2  30950  sigaclcu3  31059  fiunelros  31111  meascnbl  31156  bnj1113  31738  bnj155  31831  bnj570  31857  bnj893  31880  cvmliftlem10  32159  mrsubvrs  32322  msubvrs  32360  voliunnfl  34410  volsupnfl  34411  heiborlem3  34566  heibor  34574  iunrelexp0  39444  iunp1  40781  iundjiunlem  42202  iundjiun  42203  meaiuninclem  42223  meaiuninc  42224  carageniuncllem1  42264  carageniuncllem2  42265  carageniuncl  42266  caratheodorylem1  42269  caratheodorylem2  42270