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Theorem hboprab 5562
 Description: Bound-variable hypothesis builder for an operation class abstraction. (Contributed by set.mm contributors, 22-Aug-2013.)
Hypothesis
Ref Expression
hboprab.1 (φwφ)
Assertion
Ref Expression
hboprab (u {x, y, z φ} → w u {x, y, z φ})
Distinct variable groups:   w,u   x,w   y,w   z,w
Allowed substitution hints:   φ(x,y,z,w,u)

Proof of Theorem hboprab
Dummy variable v is distinct from all other variables.
StepHypRef Expression
1 df-oprab 5528 . 2 {x, y, z φ} = {v xyz(v = x, y, z φ)}
2 ax-17 1616 . . . . . . 7 (v = x, y, zw v = x, y, z)
3 hboprab.1 . . . . . . 7 (φwφ)
42, 3hban 1828 . . . . . 6 ((v = x, y, z φ) → w(v = x, y, z φ))
54hbex 1841 . . . . 5 (z(v = x, y, z φ) → wz(v = x, y, z φ))
65hbex 1841 . . . 4 (yz(v = x, y, z φ) → wyz(v = x, y, z φ))
76hbex 1841 . . 3 (xyz(v = x, y, z φ) → wxyz(v = x, y, z φ))
87hbab 2344 . 2 (u {v xyz(v = x, y, z φ)} → w u {v xyz(v = x, y, z φ)})
91, 8hbxfreq 2456 1 (u {x, y, z φ} → w u {x, y, z φ})
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 358  ∀wal 1540  ∃wex 1541   = wceq 1642   ∈ wcel 1710  {cab 2339  ⟨cop 4561  {coprab 5527 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-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-oprab 5528 This theorem is referenced by: (None)
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