MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  rabbia2 Structured version   Visualization version   GIF version

Theorem rabbia2 3482
Description: Equivalent wff's yield equal restricted class abstractions. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
Hypothesis
Ref Expression
rabbia2.1 ((𝑥𝐴𝜓) ↔ (𝑥𝐵𝜒))
Assertion
Ref Expression
rabbia2 {𝑥𝐴𝜓} = {𝑥𝐵𝜒}

Proof of Theorem rabbia2
StepHypRef Expression
1 rabbia2.1 . . . 4 ((𝑥𝐴𝜓) ↔ (𝑥𝐵𝜒))
21a1i 11 . . 3 (⊤ → ((𝑥𝐴𝜓) ↔ (𝑥𝐵𝜒)))
32rabbidva2 3481 . 2 (⊤ → {𝑥𝐴𝜓} = {𝑥𝐵𝜒})
43mptru 1537 1 {𝑥𝐴𝜓} = {𝑥𝐵𝜒}
Colors of variables: wff setvar class
Syntax hints:  wb 207  wa 396   = wceq 1530  wtru 1531  wcel 2107  {crab 3146
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-9 2117  ax-ext 2797
This theorem depends on definitions:  df-bi 208  df-an 397  df-tru 1533  df-ex 1774  df-sb 2063  df-clab 2804  df-cleq 2818  df-rab 3151
This theorem is referenced by:  rabswap  3493  f1ossf1o  6885  finsumvtxdg2ssteplem3  27243  clwlknf1oclwwlkn  27777  clwwlknon2x  27796  numclwwlkovh  28066  ballotlem2  31632  smflim  42915  smflim2  42942  smflimsuplem1  42956  smflimsup  42964  sprvalpwn0  43473
  Copyright terms: Public domain W3C validator