Users' Mathboxes Mathbox for BJ < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  eliminable2c Structured version   Visualization version   GIF version

Theorem eliminable2c 32036
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.)
Assertion
Ref Expression
eliminable2c ({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 eliminable2c
StepHypRef Expression
1 dfcleq 2604 1 ({𝑥𝜑} = {𝑦𝜓} ↔ ∀𝑧(𝑧 ∈ {𝑥𝜑} ↔ 𝑧 ∈ {𝑦𝜓}))
Colors of variables: wff setvar class
Syntax hints:  wb 195  wal 1473   = wceq 1475  wcel 1977  {cab 2596
This theorem was proved from axioms:  ax-ext 2590
This theorem depends on definitions:  df-cleq 2603
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator