Theorem iuneq1d 4938

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 4927	. 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 1533  ∪ ciun 4911

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rex 3144  df-v 3496  df-in 3942  df-ss 3951  df-iun 4913

This theorem is referenced by:  iuneq12d  4939  disjxiun  5055  kmlem11  9580  prmreclem4  16249  imasval  16778  iundisj  24143  iundisj2  24144  voliunlem1  24145  iunmbl  24148  volsup  24151  uniioombllem4  24181  iuninc  30306  iundisjf  30333  iundisj2f  30334  suppovss  30420  iundisjfi  30513  iundisj2fi  30514  iundisjcnt  30515  indval2  31268  sigaclcu3  31376  fiunelros  31428  meascnbl  31473  bnj1113  32052  bnj155  32146  bnj570  32172  bnj893  32195  cvmliftlem10  32536  mrsubvrs  32764  msubvrs  32802  voliunnfl  34930  volsupnfl  34931  heiborlem3  35085  heibor  35093  iunrelexp0  40040  iunp1  41321  iundjiunlem  42735  iundjiun  42736  meaiuninclem  42756  meaiuninc  42757  carageniuncllem1  42797  carageniuncllem2  42798  carageniuncl  42799  caratheodorylem1  42802  caratheodorylem2  42803