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Theorem eqrdav 2823
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 232 . . . . 5 ((𝜑𝑥𝐶) → (𝑥𝐴𝑥𝐵))
43impancom 455 . . . 4 ((𝜑𝑥𝐴) → (𝑥𝐶𝑥𝐵))
51, 4mpd 15 . . 3 ((𝜑𝑥𝐴) → 𝑥𝐵)
6 eqrdav.2 . . . 4 ((𝜑𝑥𝐵) → 𝑥𝐶)
72biimprd 251 . . . . 5 ((𝜑𝑥𝐶) → (𝑥𝐵𝑥𝐴))
87impancom 455 . . . 4 ((𝜑𝑥𝐵) → (𝑥𝐶𝑥𝐴))
96, 8mpd 15 . . 3 ((𝜑𝑥𝐵) → 𝑥𝐴)
105, 9impbida 800 . 2 (𝜑 → (𝑥𝐴𝑥𝐵))
1110eqrdv 2822 1 (𝜑𝐴 = 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1538  wcel 2115
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-9 2125  ax-ext 2796
This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-cleq 2817
This theorem is referenced by:  boxcutc  8497  supminf  12330  f1omvdconj  18572  fmucndlem  22895  lsmsnorb  30973  ballotlemsima  31800  supminfxr  41969
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