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Theorem difrab 3860
 Description: Difference of two restricted class abstractions. (Contributed by NM, 23-Oct-2004.)
Assertion
Ref Expression
difrab ({𝑥𝐴𝜑} ∖ {𝑥𝐴𝜓}) = {𝑥𝐴 ∣ (𝜑 ∧ ¬ 𝜓)}

Proof of Theorem difrab
StepHypRef Expression
1 df-rab 2905 . . 3 {𝑥𝐴𝜑} = {𝑥 ∣ (𝑥𝐴𝜑)}
2 df-rab 2905 . . 3 {𝑥𝐴𝜓} = {𝑥 ∣ (𝑥𝐴𝜓)}
31, 2difeq12i 3688 . 2 ({𝑥𝐴𝜑} ∖ {𝑥𝐴𝜓}) = ({𝑥 ∣ (𝑥𝐴𝜑)} ∖ {𝑥 ∣ (𝑥𝐴𝜓)})
4 df-rab 2905 . . 3 {𝑥𝐴 ∣ (𝜑 ∧ ¬ 𝜓)} = {𝑥 ∣ (𝑥𝐴 ∧ (𝜑 ∧ ¬ 𝜓))}
5 difab 3855 . . . 4 ({𝑥 ∣ (𝑥𝐴𝜑)} ∖ {𝑥 ∣ (𝑥𝐴𝜓)}) = {𝑥 ∣ ((𝑥𝐴𝜑) ∧ ¬ (𝑥𝐴𝜓))}
6 anass 679 . . . . . 6 (((𝑥𝐴𝜑) ∧ ¬ 𝜓) ↔ (𝑥𝐴 ∧ (𝜑 ∧ ¬ 𝜓)))
7 simpr 476 . . . . . . . . 9 ((𝑥𝐴𝜓) → 𝜓)
87con3i 149 . . . . . . . 8 𝜓 → ¬ (𝑥𝐴𝜓))
98anim2i 591 . . . . . . 7 (((𝑥𝐴𝜑) ∧ ¬ 𝜓) → ((𝑥𝐴𝜑) ∧ ¬ (𝑥𝐴𝜓)))
10 pm3.2 462 . . . . . . . . . 10 (𝑥𝐴 → (𝜓 → (𝑥𝐴𝜓)))
1110adantr 480 . . . . . . . . 9 ((𝑥𝐴𝜑) → (𝜓 → (𝑥𝐴𝜓)))
1211con3d 147 . . . . . . . 8 ((𝑥𝐴𝜑) → (¬ (𝑥𝐴𝜓) → ¬ 𝜓))
1312imdistani 722 . . . . . . 7 (((𝑥𝐴𝜑) ∧ ¬ (𝑥𝐴𝜓)) → ((𝑥𝐴𝜑) ∧ ¬ 𝜓))
149, 13impbii 198 . . . . . 6 (((𝑥𝐴𝜑) ∧ ¬ 𝜓) ↔ ((𝑥𝐴𝜑) ∧ ¬ (𝑥𝐴𝜓)))
156, 14bitr3i 265 . . . . 5 ((𝑥𝐴 ∧ (𝜑 ∧ ¬ 𝜓)) ↔ ((𝑥𝐴𝜑) ∧ ¬ (𝑥𝐴𝜓)))
1615abbii 2726 . . . 4 {𝑥 ∣ (𝑥𝐴 ∧ (𝜑 ∧ ¬ 𝜓))} = {𝑥 ∣ ((𝑥𝐴𝜑) ∧ ¬ (𝑥𝐴𝜓))}
175, 16eqtr4i 2635 . . 3 ({𝑥 ∣ (𝑥𝐴𝜑)} ∖ {𝑥 ∣ (𝑥𝐴𝜓)}) = {𝑥 ∣ (𝑥𝐴 ∧ (𝜑 ∧ ¬ 𝜓))}
184, 17eqtr4i 2635 . 2 {𝑥𝐴 ∣ (𝜑 ∧ ¬ 𝜓)} = ({𝑥 ∣ (𝑥𝐴𝜑)} ∖ {𝑥 ∣ (𝑥𝐴𝜓)})
193, 18eqtr4i 2635 1 ({𝑥𝐴𝜑} ∖ {𝑥𝐴𝜓}) = {𝑥𝐴 ∣ (𝜑 ∧ ¬ 𝜓)}
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977  {cab 2596  {crab 2900   ∖ cdif 3537 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rab 2905  df-v 3175  df-dif 3543 This theorem is referenced by:  alephsuc3  9281  shftmbl  23113  musum  24717  clwlknclwlkdifs  26487  aciunf1  28845  poimirlem26  32605  poimirlem27  32606  clwwlknclwwlkdifs  41181
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