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

Theorem eqrdav 2650
 Description: Deduce equality of classes from an equivalence of membership that depends on the membership variable. (Contributed by NM, 7-Nov-2008.) (Proof shortened by Wolf Lammen, 19-Nov-2019.)
Hypotheses
Ref Expression
eqrdav.1 ((𝜑𝑥𝐴) → 𝑥𝐶)
eqrdav.2 ((𝜑𝑥𝐵) → 𝑥𝐶)
eqrdav.3 ((𝜑𝑥𝐶) → (𝑥𝐴𝑥𝐵))
Assertion
Ref Expression
eqrdav (𝜑𝐴 = 𝐵)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝜑,𝑥
Allowed substitution hint:   𝐶(𝑥)

Proof of Theorem eqrdav
StepHypRef Expression
1 eqrdav.1 . . . 4 ((𝜑𝑥𝐴) → 𝑥𝐶)
2 eqrdav.3 . . . . . 6 ((𝜑𝑥𝐶) → (𝑥𝐴𝑥𝐵))
32biimpd 219 . . . . 5 ((𝜑𝑥𝐶) → (𝑥𝐴𝑥𝐵))
43impancom 455 . . . 4 ((𝜑𝑥𝐴) → (𝑥𝐶𝑥𝐵))
51, 4mpd 15 . . 3 ((𝜑𝑥𝐴) → 𝑥𝐵)
6 eqrdav.2 . . . 4 ((𝜑𝑥𝐵) → 𝑥𝐶)
72biimprd 238 . . . . 5 ((𝜑𝑥𝐶) → (𝑥𝐵𝑥𝐴))
87impancom 455 . . . 4 ((𝜑𝑥𝐵) → (𝑥𝐶𝑥𝐴))
96, 8mpd 15 . . 3 ((𝜑𝑥𝐵) → 𝑥𝐴)
105, 9impbida 895 . 2 (𝜑 → (𝑥𝐴𝑥𝐵))
1110eqrdv 2649 1 (𝜑𝐴 = 𝐵)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 383   = wceq 1523   ∈ wcel 2030 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-ext 2631 This theorem depends on definitions:  df-bi 197  df-an 385  df-ex 1745  df-cleq 2644 This theorem is referenced by:  boxcutc  7993  supminf  11813  f1omvdconj  17912  fmucndlem  22142  ballotlemsima  30705  supminfxr  40007
 Copyright terms: Public domain W3C validator