ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  hbab GIF version

Theorem hbab 2047
Description: Bound-variable hypothesis builder for a class abstraction. (Contributed by NM, 1-Mar-1995.)
Hypothesis
Ref Expression
hbab.1 (𝜑 → ∀𝑥𝜑)
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 2043 . 2 (𝑧 ∈ {𝑦𝜑} ↔ [𝑧 / 𝑦]𝜑)
2 hbab.1 . . 3 (𝜑 → ∀𝑥𝜑)
32hbsb 1839 . 2 ([𝑧 / 𝑦]𝜑 → ∀𝑥[𝑧 / 𝑦]𝜑)
41, 3hbxfrbi 1377 1 (𝑧 ∈ {𝑦𝜑} → ∀𝑥 𝑧 ∈ {𝑦𝜑})
Colors of variables: wff set class
Syntax hints:  wi 4  wal 1257  wcel 1409  [wsb 1661  {cab 2042
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444
This theorem depends on definitions:  df-bi 114  df-nf 1366  df-sb 1662  df-clab 2043
This theorem is referenced by:  nfsab  2048
  Copyright terms: Public domain W3C validator