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Theorem unab 3198
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 ∣ (φ ψ)}

Proof of Theorem unab
Dummy variable y is distinct from all other variables.
StepHypRef Expression
1 sbor 1825 . . 3 ([y / x](φ ψ) ↔ ([y / x]φ [y / x]ψ))
2 df-clab 2024 . . 3 (y {x ∣ (φ ψ)} ↔ [y / x](φ ψ))
3 df-clab 2024 . . . 4 (y {xφ} ↔ [y / x]φ)
4 df-clab 2024 . . . 4 (y {xψ} ↔ [y / x]ψ)
53, 4orbi12i 680 . . 3 ((y {xφ} y {xψ}) ↔ ([y / x]φ [y / x]ψ))
61, 2, 53bitr4ri 202 . 2 ((y {xφ} y {xψ}) ↔ y {x ∣ (φ ψ)})
76uneqri 3079 1 ({xφ} ∪ {xψ}) = {x ∣ (φ ψ)}
Colors of variables: wff set class
Syntax hints:   wo 628   = wceq 1242   wcel 1390  [wsb 1642  {cab 2023  cun 2909
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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553  df-un 2916
This theorem is referenced by:  unrab  3202  rabun2  3210  dfif6  3327  unopab  3827  dmun  4485  frecabex  5923
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