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Theorem hbab 2344
 Description: Bound-variable hypothesis builder for a class abstraction. (Contributed by NM, 1-Mar-1995.)
Hypothesis
Ref Expression
hbab.1 (φxφ)
Assertion
Ref Expression
hbab (z {y φ} → x z {y φ})
Distinct variable group:   x,z
Allowed substitution hints:   φ(x,y,z)

Proof of Theorem hbab
StepHypRef Expression
1 df-clab 2340 . 2 (z {y φ} ↔ [z / y]φ)
2 hbab.1 . . 3 (φxφ)
32hbsb 2110 . 2 ([z / y]φx[z / y]φ)
41, 3hbxfrbi 1568 1 (z {y φ} → x z {y φ})
 Colors of variables: wff setvar class Syntax hints:   → wi 4  ∀wal 1540  [wsb 1648   ∈ wcel 1710  {cab 2339 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 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 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 This theorem is referenced by:  nfsab  2345  hboprab1  5559  hboprab2  5560  hboprab3  5561  hboprab  5562
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