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Theorem ssconb 3399
 Description: Contraposition law for subsets. (Contributed by NM, 22-Mar-1998.)
Assertion
Ref Expression
ssconb ((A C B C) → (A (C B) ↔ B (C A)))

Proof of Theorem ssconb
Dummy variable x is distinct from all other variables.
StepHypRef Expression
1 ssel 3267 . . . . . . 7 (A C → (x Ax C))
2 ssel 3267 . . . . . . 7 (B C → (x Bx C))
3 pm5.1 830 . . . . . . 7 (((x Ax C) (x Bx C)) → ((x Ax C) ↔ (x Bx C)))
41, 2, 3syl2an 463 . . . . . 6 ((A C B C) → ((x Ax C) ↔ (x Bx C)))
5 con2b 324 . . . . . . 7 ((x A → ¬ x B) ↔ (x B → ¬ x A))
65a1i 10 . . . . . 6 ((A C B C) → ((x A → ¬ x B) ↔ (x B → ¬ x A)))
74, 6anbi12d 691 . . . . 5 ((A C B C) → (((x Ax C) (x A → ¬ x B)) ↔ ((x Bx C) (x B → ¬ x A))))
8 jcab 833 . . . . 5 ((x A → (x C ¬ x B)) ↔ ((x Ax C) (x A → ¬ x B)))
9 jcab 833 . . . . 5 ((x B → (x C ¬ x A)) ↔ ((x Bx C) (x B → ¬ x A)))
107, 8, 93bitr4g 279 . . . 4 ((A C B C) → ((x A → (x C ¬ x B)) ↔ (x B → (x C ¬ x A))))
11 eldif 3221 . . . . 5 (x (C B) ↔ (x C ¬ x B))
1211imbi2i 303 . . . 4 ((x Ax (C B)) ↔ (x A → (x C ¬ x B)))
13 eldif 3221 . . . . 5 (x (C A) ↔ (x C ¬ x A))
1413imbi2i 303 . . . 4 ((x Bx (C A)) ↔ (x B → (x C ¬ x A)))
1510, 12, 143bitr4g 279 . . 3 ((A C B C) → ((x Ax (C B)) ↔ (x Bx (C A))))
1615albidv 1625 . 2 ((A C B C) → (x(x Ax (C B)) ↔ x(x Bx (C A))))
17 dfss2 3262 . 2 (A (C B) ↔ x(x Ax (C B)))
18 dfss2 3262 . 2 (B (C A) ↔ x(x Bx (C A)))
1916, 17, 183bitr4g 279 1 ((A C B C) → (A (C B) ↔ B (C A)))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 176   ∧ wa 358  ∀wal 1540   ∈ wcel 1710   ∖ cdif 3206   ⊆ 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:  pssdifcom1  3635  pssdifcom2  3636
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