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Theorem eqrdav 2352
 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 198 . . . . 5 ((φ x C) → (x Ax B))
43impancom 427 . . . 4 ((φ x A) → (x Cx B))
51, 4mpd 14 . . 3 ((φ x A) → x B)
6 eqrdav.2 . . . 4 ((φ x B) → x C)
72exbiri 605 . . . . . 6 (φ → (x C → (x Bx A)))
87com23 72 . . . . 5 (φ → (x B → (x Cx A)))
98imp 418 . . . 4 ((φ x B) → (x Cx A))
106, 9mpd 14 . . 3 ((φ x B) → x A)
115, 10impbida 805 . 2 (φ → (x Ax B))
1211eqrdv 2351 1 (φA = B)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358   = wceq 1642   ∈ wcel 1710 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-an 360  df-cleq 2346 This theorem is referenced by: (None)
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