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Theorem difrab 3529
 Description: Difference of two restricted class abstractions. (Contributed by NM, 23-Oct-2004.)
Assertion
Ref Expression
difrab ({x A φ} {x A ψ}) = {x A (φ ¬ ψ)}

Proof of Theorem difrab
StepHypRef Expression
1 df-rab 2623 . . 3 {x A φ} = {x (x A φ)}
2 df-rab 2623 . . 3 {x A ψ} = {x (x A ψ)}
31, 2difeq12i 3383 . 2 ({x A φ} {x A ψ}) = ({x (x A φ)} {x (x A ψ)})
4 df-rab 2623 . . 3 {x A (φ ¬ ψ)} = {x (x A (φ ¬ ψ))}
5 difab 3523 . . . 4 ({x (x A φ)} {x (x A ψ)}) = {x ((x A φ) ¬ (x A ψ))}
6 anass 630 . . . . . 6 (((x A φ) ¬ ψ) ↔ (x A (φ ¬ ψ)))
7 simpr 447 . . . . . . . . 9 ((x A ψ) → ψ)
87con3i 127 . . . . . . . 8 ψ → ¬ (x A ψ))
98anim2i 552 . . . . . . 7 (((x A φ) ¬ ψ) → ((x A φ) ¬ (x A ψ)))
10 pm3.2 434 . . . . . . . . . 10 (x A → (ψ → (x A ψ)))
1110adantr 451 . . . . . . . . 9 ((x A φ) → (ψ → (x A ψ)))
1211con3d 125 . . . . . . . 8 ((x A φ) → (¬ (x A ψ) → ¬ ψ))
1312imdistani 671 . . . . . . 7 (((x A φ) ¬ (x A ψ)) → ((x A φ) ¬ ψ))
149, 13impbii 180 . . . . . 6 (((x A φ) ¬ ψ) ↔ ((x A φ) ¬ (x A ψ)))
156, 14bitr3i 242 . . . . 5 ((x A (φ ¬ ψ)) ↔ ((x A φ) ¬ (x A ψ)))
1615abbii 2465 . . . 4 {x (x A (φ ¬ ψ))} = {x ((x A φ) ¬ (x A ψ))}
175, 16eqtr4i 2376 . . 3 ({x (x A φ)} {x (x A ψ)}) = {x (x A (φ ¬ ψ))}
184, 17eqtr4i 2376 . 2 {x A (φ ¬ ψ)} = ({x (x A φ)} {x (x A ψ)})
193, 18eqtr4i 2376 1 ({x A φ} {x A ψ}) = {x A (φ ¬ ψ)}
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 358   = wceq 1642   ∈ wcel 1710  {cab 2339  {crab 2618   ∖ cdif 3206 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-rab 2623  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-dif 3215 This theorem is referenced by: (None)
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