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

Theorem abbid 2808
Description: Equivalent wff's yield equal class abstractions (deduction form, with nonfreeness hypothesis). (Contributed by NM, 21-Jun-1993.) (Revised by Mario Carneiro, 7-Oct-2016.) Avoid ax-10 2139 and ax-11 2155. (Revised by Wolf Lammen, 6-May-2023.)
Hypotheses
Ref Expression
abbid.1 𝑥𝜑
abbid.2 (𝜑 → (𝜓𝜒))
Assertion
Ref Expression
abbid (𝜑 → {𝑥𝜓} = {𝑥𝜒})

Proof of Theorem abbid
StepHypRef Expression
1 abbid.1 . . 3 𝑥𝜑
2 abbid.2 . . 3 (𝜑 → (𝜓𝜒))
31, 2alrimi 2211 . 2 (𝜑 → ∀𝑥(𝜓𝜒))
4 abbi 2805 . 2 (∀𝑥(𝜓𝜒) → {𝑥𝜓} = {𝑥𝜒})
53, 4syl 17 1 (𝜑 → {𝑥𝜓} = {𝑥𝜒})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wal 1535   = wceq 1537  wnf 1780  {cab 2712
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-9 2116  ax-12 2175  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1777  df-nf 1781  df-sb 2063  df-clab 2713  df-cleq 2727
This theorem is referenced by:  rabbida4  3460  rabeqf  3470  sbcbid  3850  sbceqbidf  32515  opabdm  32631  opabrn  32632  fpwrelmap  32751  sticksstones16  42144  rabbida2  45072  rabbida3  45075
  Copyright terms: Public domain W3C validator