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

Theorem abbid 2737
 Description: Equivalent wff's yield equal class abstractions (deduction rule). (Contributed by NM, 21-Jun-1993.) (Revised by Mario Carneiro, 7-Oct-2016.)
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 2080 . 2 (𝜑 → ∀𝑥(𝜓𝜒))
4 abbi 2734 . 2 (∀𝑥(𝜓𝜒) ↔ {𝑥𝜓} = {𝑥𝜒})
53, 4sylib 208 1 (𝜑 → {𝑥𝜓} = {𝑥𝜒})
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196  ∀wal 1478   = wceq 1480  Ⅎwnf 1705  {cab 2607 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617 This theorem is referenced by:  abbidv  2738  rabeqf  3178  sbcbid  3471  sbceqbidf  29167  opabdm  29263  opabrn  29264  fpwrelmap  29348  bj-rabbida2  32557  iotain  38097  rabbida2  38803  rabbida3  38806
 Copyright terms: Public domain W3C validator