Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  unab Structured version   Visualization version   GIF version

Theorem unab 4255
 Description: Union of two class abstractions. (Contributed by NM, 29-Sep-2002.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
Assertion
Ref Expression
unab ({𝑥𝜑} ∪ {𝑥𝜓}) = {𝑥 ∣ (𝜑𝜓)}

Proof of Theorem unab
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 sbor 2318 . . 3 ([𝑦 / 𝑥](𝜑𝜓) ↔ ([𝑦 / 𝑥]𝜑 ∨ [𝑦 / 𝑥]𝜓))
2 df-clab 2803 . . 3 (𝑦 ∈ {𝑥 ∣ (𝜑𝜓)} ↔ [𝑦 / 𝑥](𝜑𝜓))
3 df-clab 2803 . . . 4 (𝑦 ∈ {𝑥𝜑} ↔ [𝑦 / 𝑥]𝜑)
4 df-clab 2803 . . . 4 (𝑦 ∈ {𝑥𝜓} ↔ [𝑦 / 𝑥]𝜓)
53, 4orbi12i 912 . . 3 ((𝑦 ∈ {𝑥𝜑} ∨ 𝑦 ∈ {𝑥𝜓}) ↔ ([𝑦 / 𝑥]𝜑 ∨ [𝑦 / 𝑥]𝜓))
61, 2, 53bitr4ri 307 . 2 ((𝑦 ∈ {𝑥𝜑} ∨ 𝑦 ∈ {𝑥𝜓}) ↔ 𝑦 ∈ {𝑥 ∣ (𝜑𝜓)})
76uneqri 4113 1 ({𝑥𝜑} ∪ {𝑥𝜓}) = {𝑥 ∣ (𝜑𝜓)}
 Colors of variables: wff setvar class Syntax hints:   ∨ wo 844   = wceq 1538  [wsb 2070   ∈ wcel 2115  {cab 2802   ∪ cun 3917 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 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-12 2179  ax-ext 2796 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-ex 1782  df-nf 1786  df-sb 2071  df-clab 2803  df-cleq 2817  df-clel 2896  df-v 3482  df-un 3924 This theorem is referenced by:  unrab  4259  rabun2  4267  dfif6  4453  unopab  5131  dmun  5766  hashf1lem2  13819  vdwlem6  16320  vtxdun  27274  satfvsuclem1  32663  satf0suclem  32679  fmlasuc0  32688  diophun  39630
 Copyright terms: Public domain W3C validator