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Theorem difrab 4214
 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 3080 . . 3 {𝑥𝐴𝜑} = {𝑥 ∣ (𝑥𝐴𝜑)}
2 df-rab 3080 . . 3 {𝑥𝐴𝜓} = {𝑥 ∣ (𝑥𝐴𝜓)}
31, 2difeq12i 4029 . 2 ({𝑥𝐴𝜑} ∖ {𝑥𝐴𝜓}) = ({𝑥 ∣ (𝑥𝐴𝜑)} ∖ {𝑥 ∣ (𝑥𝐴𝜓)})
4 df-rab 3080 . . 3 {𝑥𝐴 ∣ (𝜑 ∧ ¬ 𝜓)} = {𝑥 ∣ (𝑥𝐴 ∧ (𝜑 ∧ ¬ 𝜓))}
5 difab 4207 . . . 4 ({𝑥 ∣ (𝑥𝐴𝜑)} ∖ {𝑥 ∣ (𝑥𝐴𝜓)}) = {𝑥 ∣ ((𝑥𝐴𝜑) ∧ ¬ (𝑥𝐴𝜓))}
6 anass 472 . . . . . 6 (((𝑥𝐴𝜑) ∧ ¬ 𝜓) ↔ (𝑥𝐴 ∧ (𝜑 ∧ ¬ 𝜓)))
7 simpr 488 . . . . . . . . 9 ((𝑥𝐴𝜓) → 𝜓)
87con3i 157 . . . . . . . 8 𝜓 → ¬ (𝑥𝐴𝜓))
98anim2i 619 . . . . . . 7 (((𝑥𝐴𝜑) ∧ ¬ 𝜓) → ((𝑥𝐴𝜑) ∧ ¬ (𝑥𝐴𝜓)))
10 pm3.2 473 . . . . . . . . . 10 (𝑥𝐴 → (𝜓 → (𝑥𝐴𝜓)))
1110adantr 484 . . . . . . . . 9 ((𝑥𝐴𝜑) → (𝜓 → (𝑥𝐴𝜓)))
1211con3d 155 . . . . . . . 8 ((𝑥𝐴𝜑) → (¬ (𝑥𝐴𝜓) → ¬ 𝜓))
1312imdistani 572 . . . . . . 7 (((𝑥𝐴𝜑) ∧ ¬ (𝑥𝐴𝜓)) → ((𝑥𝐴𝜑) ∧ ¬ 𝜓))
149, 13impbii 212 . . . . . 6 (((𝑥𝐴𝜑) ∧ ¬ 𝜓) ↔ ((𝑥𝐴𝜑) ∧ ¬ (𝑥𝐴𝜓)))
156, 14bitr3i 280 . . . . 5 ((𝑥𝐴 ∧ (𝜑 ∧ ¬ 𝜓)) ↔ ((𝑥𝐴𝜑) ∧ ¬ (𝑥𝐴𝜓)))
1615abbii 2824 . . . 4 {𝑥 ∣ (𝑥𝐴 ∧ (𝜑 ∧ ¬ 𝜓))} = {𝑥 ∣ ((𝑥𝐴𝜑) ∧ ¬ (𝑥𝐴𝜓))}
175, 16eqtr4i 2785 . . 3 ({𝑥 ∣ (𝑥𝐴𝜑)} ∖ {𝑥 ∣ (𝑥𝐴𝜓)}) = {𝑥 ∣ (𝑥𝐴 ∧ (𝜑 ∧ ¬ 𝜓))}
184, 17eqtr4i 2785 . 2 {𝑥𝐴 ∣ (𝜑 ∧ ¬ 𝜓)} = ({𝑥 ∣ (𝑥𝐴𝜑)} ∖ {𝑥 ∣ (𝑥𝐴𝜓)})
193, 18eqtr4i 2785 1 ({𝑥𝐴𝜑} ∖ {𝑥𝐴𝜓}) = {𝑥𝐴 ∣ (𝜑 ∧ ¬ 𝜓)}
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 399   = wceq 1539   ∈ wcel 2112  {cab 2736  {crab 3075   ∖ cdif 3858 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2114  ax-9 2122  ax-10 2143  ax-12 2176  ax-ext 2730 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1542  df-ex 1783  df-nf 1787  df-sb 2071  df-clab 2737  df-cleq 2751  df-clel 2831  df-rab 3080  df-v 3412  df-dif 3864 This theorem is referenced by:  alephsuc3  10054  shftmbl  24253  musum  25890  clwwlknclwwlkdif  27878  aciunf1  30538  poimirlem26  35399  poimirlem27  35400
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