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Theorem unab 3521
Description: Union of two class abstractions. (Contributed by NM, 29-Sep-2002.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
Assertion
Ref Expression
unab ({x φ} ∪ {x ψ}) = {x (φ ψ)}

Proof of Theorem unab
Dummy variable y is distinct from all other variables.
StepHypRef Expression
1 sbor 2066 . . 3 ([y / x](φ ψ) ↔ ([y / x]φ [y / x]ψ))
2 df-clab 2340 . . 3 (y {x (φ ψ)} ↔ [y / x](φ ψ))
3 df-clab 2340 . . . 4 (y {x φ} ↔ [y / x]φ)
4 df-clab 2340 . . . 4 (y {x ψ} ↔ [y / x]ψ)
53, 4orbi12i 507 . . 3 ((y {x φ} y {x ψ}) ↔ ([y / x]φ [y / x]ψ))
61, 2, 53bitr4ri 269 . 2 ((y {x φ} y {x ψ}) ↔ y {x (φ ψ)})
76uneqri 3406 1 ({x φ} ∪ {x ψ}) = {x (φ ψ)}
Colors of variables: wff setvar class
Syntax hints:   wo 357   = wceq 1642  [wsb 1648   wcel 1710  {cab 2339  cun 3207
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-v 2861  df-nin 3211  df-compl 3212  df-un 3214
This theorem is referenced by:  unrab  3526  rabun2  3534  dfif6  3665  nnc0suc  4412  nncaddccl  4419  preaddccan2lem1  4454  ltfintrilem1  4465  nnadjoin  4520  tfinnn  4534  phiun  4614  unopab  4638  clos1basesuc  5882  leconnnc  6218  addccan2nclem2  6264  nchoicelem16  6304
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