MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  hbab Structured version   Visualization version   GIF version

Theorem hbab 2743
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 2739 . 2 (𝑧 ∈ {𝑦𝜑} ↔ [𝑧 / 𝑦]𝜑)
2 hbab.1 . . 3 (𝜑 → ∀𝑥𝜑)
32hbsb 2570 . 2 ([𝑧 / 𝑦]𝜑 → ∀𝑥[𝑧 / 𝑦]𝜑)
41, 3hbxfrbi 1893 1 (𝑧 ∈ {𝑦𝜑} → ∀𝑥 𝑧 ∈ {𝑦𝜑})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1622  [wsb 2038  wcel 2131  {cab 2738
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1863  ax-4 1878  ax-5 1980  ax-6 2046  ax-7 2082  ax-10 2160  ax-11 2175  ax-12 2188  ax-13 2383
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1627  df-ex 1846  df-nf 1851  df-sb 2039  df-clab 2739
This theorem is referenced by:  nfsab  2744  bnj1441  31210  bnj1309  31389
  Copyright terms: Public domain W3C validator