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Theorem dfss4 3489
 Description: Subclass defined in terms of class difference. See comments under dfun2 3490. (Contributed by NM, 22-Mar-1998.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
Assertion
Ref Expression
dfss4 (A B ↔ (B (B A)) = A)

Proof of Theorem dfss4
Dummy variable x is distinct from all other variables.
StepHypRef Expression
1 sseqin2 3474 . 2 (A B ↔ (BA) = A)
2 eldif 3221 . . . . . . 7 (x (B A) ↔ (x B ¬ x A))
32notbii 287 . . . . . 6 x (B A) ↔ ¬ (x B ¬ x A))
43anbi2i 675 . . . . 5 ((x B ¬ x (B A)) ↔ (x B ¬ (x B ¬ x A)))
5 elin 3219 . . . . . 6 (x (BA) ↔ (x B x A))
6 abai 770 . . . . . 6 ((x B x A) ↔ (x B (x Bx A)))
7 iman 413 . . . . . . 7 ((x Bx A) ↔ ¬ (x B ¬ x A))
87anbi2i 675 . . . . . 6 ((x B (x Bx A)) ↔ (x B ¬ (x B ¬ x A)))
95, 6, 83bitri 262 . . . . 5 (x (BA) ↔ (x B ¬ (x B ¬ x A)))
104, 9bitr4i 243 . . . 4 ((x B ¬ x (B A)) ↔ x (BA))
1110difeqri 3387 . . 3 (B (B A)) = (BA)
1211eqeq1i 2360 . 2 ((B (B A)) = A ↔ (BA) = A)
131, 12bitr4i 243 1 (A B ↔ (B (B A)) = A)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 176   ∧ wa 358   = wceq 1642   ∈ wcel 1710   ∖ cdif 3206   ∩ cin 3208   ⊆ wss 3257 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-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-dif 3215  df-ss 3259 This theorem is referenced by:  dfin4  3495
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