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Theorem eqrdav 2055
Description: Deduce equality of classes from an equivalence of membership that depends on the membership variable. (Contributed by NM, 7-Nov-2008.)
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 136 . . . . 5 ((𝜑𝑥𝐶) → (𝑥𝐴𝑥𝐵))
43impancom 251 . . . 4 ((𝜑𝑥𝐴) → (𝑥𝐶𝑥𝐵))
51, 4mpd 13 . . 3 ((𝜑𝑥𝐴) → 𝑥𝐵)
6 eqrdav.2 . . . 4 ((𝜑𝑥𝐵) → 𝑥𝐶)
72exbiri 368 . . . . . 6 (𝜑 → (𝑥𝐶 → (𝑥𝐵𝑥𝐴)))
87com23 76 . . . . 5 (𝜑 → (𝑥𝐵 → (𝑥𝐶𝑥𝐴)))
98imp 119 . . . 4 ((𝜑𝑥𝐵) → (𝑥𝐶𝑥𝐴))
106, 9mpd 13 . . 3 ((𝜑𝑥𝐵) → 𝑥𝐴)
115, 10impbida 538 . 2 (𝜑 → (𝑥𝐴𝑥𝐵))
1211eqrdv 2054 1 (𝜑𝐴 = 𝐵)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101  wb 102   = wceq 1259  wcel 1409
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-5 1352  ax-gen 1354  ax-17 1435  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-cleq 2049
This theorem is referenced by:  fzdifsuc  9044
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