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Theorem cleqh 2450
 Description: Establish equality between classes, using bound-variable hypotheses instead of distinct variable conditions. (Contributed by NM, 5-Aug-1993.)
Hypotheses
Ref Expression
cleqh.1 (y Ax y A)
cleqh.2 (y Bx y B)
Assertion
Ref Expression
cleqh (A = Bx(x Ax B))
Distinct variable groups:   y,A   y,B   x,y
Allowed substitution hints:   A(x)   B(x)

Proof of Theorem cleqh
StepHypRef Expression
1 dfcleq 2347 . 2 (A = By(y Ay B))
2 ax-17 1616 . . . 4 ((x Ax B) → y(x Ax B))
3 dfbi2 609 . . . . 5 ((y Ay B) ↔ ((y Ay B) (y By A)))
4 cleqh.1 . . . . . . 7 (y Ax y A)
5 cleqh.2 . . . . . . 7 (y Bx y B)
64, 5hbim 1817 . . . . . 6 ((y Ay B) → x(y Ay B))
75, 4hbim 1817 . . . . . 6 ((y By A) → x(y By A))
86, 7hban 1828 . . . . 5 (((y Ay B) (y By A)) → x((y Ay B) (y By A)))
93, 8hbxfrbi 1568 . . . 4 ((y Ay B) → x(y Ay B))
10 eleq1 2413 . . . . . 6 (x = y → (x Ay A))
11 eleq1 2413 . . . . . 6 (x = y → (x By B))
1210, 11bibi12d 312 . . . . 5 (x = y → ((x Ax B) ↔ (y Ay B)))
1312biimpd 198 . . . 4 (x = y → ((x Ax B) → (y Ay B)))
142, 9, 13cbv3h 1983 . . 3 (x(x Ax B) → y(y Ay B))
1512equcoms 1681 . . . . 5 (y = x → ((x Ax B) ↔ (y Ay B)))
1615biimprd 214 . . . 4 (y = x → ((y Ay B) → (x Ax B)))
179, 2, 16cbv3h 1983 . . 3 (y(y Ay B) → x(x Ax B))
1814, 17impbii 180 . 2 (x(x Ax B) ↔ y(y Ay B))
191, 18bitr4i 243 1 (A = Bx(x Ax B))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358  ∀wal 1540   = wceq 1642   ∈ wcel 1710 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-cleq 2346  df-clel 2349 This theorem is referenced by:  abeq2  2458
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