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Theorem hbab 2028
 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 {yφ} → x z {yφ})
Distinct variable group:   x,z
Allowed substitution hints:   φ(x,y,z)

Proof of Theorem hbab
StepHypRef Expression
1 df-clab 2024 . 2 (z {yφ} ↔ [z / y]φ)
2 hbab.1 . . 3 (φxφ)
32hbsb 1820 . 2 ([z / y]φx[z / y]φ)
41, 3hbxfrbi 1358 1 (z {yφ} → x z {yφ})
 Colors of variables: wff set class Syntax hints:   → wi 4  ∀wal 1240   ∈ wcel 1390  [wsb 1642  {cab 2023 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 This theorem depends on definitions:  df-bi 110  df-nf 1347  df-sb 1643  df-clab 2024 This theorem is referenced by:  nfsab  2029
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