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Theorem dfss2f 3264
 Description: Equivalence for subclass relation, using bound-variable hypotheses instead of distinct variable conditions. (Contributed by NM, 3-Jul-1994.) (Revised by Andrew Salmon, 27-Aug-2011.)
Hypotheses
Ref Expression
dfss2f.1 xA
dfss2f.2 xB
Assertion
Ref Expression
dfss2f (A Bx(x Ax B))

Proof of Theorem dfss2f
Dummy variable z is distinct from all other variables.
StepHypRef Expression
1 dfss2 3262 . 2 (A Bz(z Az B))
2 dfss2f.1 . . . . 5 xA
32nfcri 2483 . . . 4 x z A
4 dfss2f.2 . . . . 5 xB
54nfcri 2483 . . . 4 x z B
63, 5nfim 1813 . . 3 x(z Az B)
7 nfv 1619 . . 3 z(x Ax B)
8 eleq1 2413 . . . 4 (z = x → (z Ax A))
9 eleq1 2413 . . . 4 (z = x → (z Bx B))
108, 9imbi12d 311 . . 3 (z = x → ((z Az B) ↔ (x Ax B)))
116, 7, 10cbval 1984 . 2 (z(z Az B) ↔ x(x Ax B))
121, 11bitri 240 1 (A Bx(x Ax B))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176  ∀wal 1540   = wceq 1642   ∈ wcel 1710  Ⅎwnfc 2476   ⊆ wss 3257 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-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-ss 3259 This theorem is referenced by:  dfss3f  3265  ss2ab  3334
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