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Theorem unab 3204
 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 1828 . . 3 ([𝑦 / 𝑥](𝜑𝜓) ↔ ([𝑦 / 𝑥]𝜑 ∨ [𝑦 / 𝑥]𝜓))
2 df-clab 2027 . . 3 (𝑦 ∈ {𝑥 ∣ (𝜑𝜓)} ↔ [𝑦 / 𝑥](𝜑𝜓))
3 df-clab 2027 . . . 4 (𝑦 ∈ {𝑥𝜑} ↔ [𝑦 / 𝑥]𝜑)
4 df-clab 2027 . . . 4 (𝑦 ∈ {𝑥𝜓} ↔ [𝑦 / 𝑥]𝜓)
53, 4orbi12i 681 . . 3 ((𝑦 ∈ {𝑥𝜑} ∨ 𝑦 ∈ {𝑥𝜓}) ↔ ([𝑦 / 𝑥]𝜑 ∨ [𝑦 / 𝑥]𝜓))
61, 2, 53bitr4ri 202 . 2 ((𝑦 ∈ {𝑥𝜑} ∨ 𝑦 ∈ {𝑥𝜓}) ↔ 𝑦 ∈ {𝑥 ∣ (𝜑𝜓)})
76uneqri 3085 1 ({𝑥𝜑} ∪ {𝑥𝜓}) = {𝑥 ∣ (𝜑𝜓)}
 Colors of variables: wff set class Syntax hints:   ∨ wo 629   = wceq 1243   ∈ wcel 1393  [wsb 1645  {cab 2026   ∪ cun 2915 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022 This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-un 2922 This theorem is referenced by:  unrab  3208  rabun2  3216  dfif6  3333  unopab  3836  dmun  4542  frecabex  5984
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