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Theorem cleqh 2120
Description: Establish equality between classes, using bound-variable hypotheses instead of distinct variable conditions. See also cleqf 2184. (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 2017 . 2 (A = By(y Ay B))
2 ax-17 1401 . . . 4 ((x Ax B) → y(x Ax B))
3 dfbi2 368 . . . . 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 1420 . . . . . 6 ((y Ay B) → x(y Ay B))
75, 4hbim 1420 . . . . . 6 ((y By A) → x(y By A))
86, 7hban 1422 . . . . 5 (((y Ay B) (y By A)) → x((y Ay B) (y By A)))
93, 8hbxfrbi 1341 . . . 4 ((y Ay B) → x(y Ay B))
10 eleq1 2083 . . . . . 6 (x = y → (x Ay A))
11 eleq1 2083 . . . . . 6 (x = y → (x By B))
1210, 11bibi12d 224 . . . . 5 (x = y → ((x Ax B) ↔ (y Ay B)))
1312biimpd 132 . . . 4 (x = y → ((x Ax B) → (y Ay B)))
142, 9, 13cbv3h 1614 . . 3 (x(x Ax B) → y(y Ay B))
1512equcoms 1577 . . . . 5 (y = x → ((x Ax B) ↔ (y Ay B)))
1615biimprd 147 . . . 4 (y = x → ((y Ay B) → (x Ax B)))
179, 2, 16cbv3h 1614 . . 3 (y(y Ay B) → x(x Ax B))
1814, 17impbii 117 . 2 (x(x Ax B) ↔ y(y Ay B))
191, 18bitr4i 176 1 (A = Bx(x Ax B))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98  wal 1226   = wceq 1228   wcel 1375
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 1316  ax-7 1317  ax-gen 1318  ax-ie1 1364  ax-ie2 1365  ax-8 1377  ax-4 1382  ax-17 1401  ax-i9 1405  ax-ial 1410  ax-i5r 1411  ax-ext 2005
This theorem depends on definitions:  df-bi 110  df-nf 1330  df-cleq 2016  df-clel 2019
This theorem is referenced by:  abeq2  2129
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