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Theorem unab 3232
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 1844 . . 3 ([𝑦 / 𝑥](𝜑𝜓) ↔ ([𝑦 / 𝑥]𝜑 ∨ [𝑦 / 𝑥]𝜓))
2 df-clab 2043 . . 3 (𝑦 ∈ {𝑥 ∣ (𝜑𝜓)} ↔ [𝑦 / 𝑥](𝜑𝜓))
3 df-clab 2043 . . . 4 (𝑦 ∈ {𝑥𝜑} ↔ [𝑦 / 𝑥]𝜑)
4 df-clab 2043 . . . 4 (𝑦 ∈ {𝑥𝜓} ↔ [𝑦 / 𝑥]𝜓)
53, 4orbi12i 691 . . 3 ((𝑦 ∈ {𝑥𝜑} ∨ 𝑦 ∈ {𝑥𝜓}) ↔ ([𝑦 / 𝑥]𝜑 ∨ [𝑦 / 𝑥]𝜓))
61, 2, 53bitr4ri 206 . 2 ((𝑦 ∈ {𝑥𝜑} ∨ 𝑦 ∈ {𝑥𝜓}) ↔ 𝑦 ∈ {𝑥 ∣ (𝜑𝜓)})
76uneqri 3113 1 ({𝑥𝜑} ∪ {𝑥𝜓}) = {𝑥 ∣ (𝜑𝜓)}
Colors of variables: wff set class
Syntax hints:  wo 639   = wceq 1259  wcel 1409  [wsb 1661  {cab 2042  cun 2943
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-v 2576  df-un 2950
This theorem is referenced by:  unrab  3236  rabun2  3244  dfif6  3361  unopab  3864  dmun  4570  frecabex  6015
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