Users' Mathboxes Mathbox for BJ < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  eliminable3b Structured version   Visualization version   GIF version

Theorem eliminable3b 33179
Description: A theorem used to prove the base case of the Eliminability Theorem (see section comment). (Contributed by BJ, 19-Oct-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
Ref Expression
eliminable3b ({𝑥𝜑} ∈ {𝑦𝜓} ↔ ∃𝑧(𝑧 = {𝑥𝜑} ∧ 𝑧 ∈ {𝑦𝜓}))
Distinct variable groups:   𝑥,𝑧   𝑦,𝑧   𝜑,𝑧   𝜓,𝑧
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑥,𝑦)

Proof of Theorem eliminable3b
StepHypRef Expression
1 df-clel 2767 1 ({𝑥𝜑} ∈ {𝑦𝜓} ↔ ∃𝑧(𝑧 = {𝑥𝜑} ∧ 𝑧 ∈ {𝑦𝜓}))
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 382   = wceq 1631  wex 1852  wcel 2145  {cab 2757
This theorem depends on definitions:  df-clel 2767
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator