Theorem iuneq1d 4964

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 4953	. 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 1541  ∪ ciun 4936

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 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1544  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-rex 3057  df-v 3438  df-ss 3914  df-iun 4938

This theorem is referenced by:  iuneq12dOLD  4965  disjxiun  5083  kmlem11  10047  prmreclem4  16826  imasval  17410  iundisj  25471  iundisj2  25472  voliunlem1  25473  iunmbl  25476  volsup  25479  uniioombllem4  25509  iuninc  32532  iundisjf  32561  iundisj2f  32562  suppovss  32654  iundisjfi  32770  iundisj2fi  32771  iundisjcnt  32772  indval2  32827  sigaclcu3  34127  fiunelros  34179  meascnbl  34224  bnj1113  34789  bnj155  34883  bnj570  34909  bnj893  34932  cvmliftlem10  35330  mrsubvrs  35558  msubvrs  35596  voliunnfl  37704  volsupnfl  37705  heiborlem3  37853  heibor  37861  iunrelexp0  43735  iunp1  45103  iundjiunlem  46497  iundjiun  46498  meaiuninclem  46518  meaiuninc  46519  carageniuncllem1  46559  carageniuncllem2  46560  carageniuncl  46561  caratheodorylem1  46564  caratheodorylem2  46565  imasubclem3  49138  imaf1hom  49140