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Theorem eqrdav 2036
 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 ((φ x A) → x 𝐶)
eqrdav.2 ((φ x B) → x 𝐶)
eqrdav.3 ((φ x 𝐶) → (x Ax B))
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 ((φ x A) → x 𝐶)
2 eqrdav.3 . . . . . 6 ((φ x 𝐶) → (x Ax B))
32biimpd 132 . . . . 5 ((φ x 𝐶) → (x Ax B))
43impancom 247 . . . 4 ((φ x A) → (x 𝐶x B))
51, 4mpd 13 . . 3 ((φ x A) → x B)
6 eqrdav.2 . . . 4 ((φ x B) → x 𝐶)
72exbiri 364 . . . . . 6 (φ → (x 𝐶 → (x Bx A)))
87com23 72 . . . . 5 (φ → (x B → (x 𝐶x A)))
98imp 115 . . . 4 ((φ x B) → (x 𝐶x A))
106, 9mpd 13 . . 3 ((φ x B) → x A)
115, 10impbida 528 . 2 (φ → (x Ax B))
1211eqrdv 2035 1 (φA = B)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98   = wceq 1242   ∈ wcel 1390 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1333  ax-gen 1335  ax-17 1416  ax-ext 2019 This theorem depends on definitions:  df-bi 110  df-cleq 2030 This theorem is referenced by:  fzdifsuc  8713
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