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

Theorem eqri 3990
 Description: Infer equality of classes from equivalence of membership. (Contributed by Thierry Arnoux, 7-Oct-2017.)
Hypotheses
Ref Expression
eqri.1 𝑥𝐴
eqri.2 𝑥𝐵
eqri.3 (𝑥𝐴𝑥𝐵)
Assertion
Ref Expression
eqri 𝐴 = 𝐵

Proof of Theorem eqri
StepHypRef Expression
1 nftru 1798 . . 3 𝑥
2 eqri.1 . . 3 𝑥𝐴
3 eqri.2 . . 3 𝑥𝐵
4 eqri.3 . . . 4 (𝑥𝐴𝑥𝐵)
54a1i 11 . . 3 (⊤ → (𝑥𝐴𝑥𝐵))
61, 2, 3, 5eqrd 3989 . 2 (⊤ → 𝐴 = 𝐵)
76mptru 1537 1 𝐴 = 𝐵
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 207   = wceq 1530  ⊤wtru 1531   ∈ wcel 2107  Ⅎwnfc 2965 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-8 2109  ax-9 2117  ax-11 2153  ax-12 2169  ax-ext 2797 This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-tru 1533  df-ex 1774  df-nf 1778  df-cleq 2818  df-clel 2897  df-nfc 2967 This theorem is referenced by:  rnep  5795  difrab2  30178  esum2dlem  31240  eulerpartlemn  31528
 Copyright terms: Public domain W3C validator