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Theorem difrab 3879
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 2916 . . 3 {𝑥𝐴𝜑} = {𝑥 ∣ (𝑥𝐴𝜑)}
2 df-rab 2916 . . 3 {𝑥𝐴𝜓} = {𝑥 ∣ (𝑥𝐴𝜓)}
31, 2difeq12i 3706 . 2 ({𝑥𝐴𝜑} ∖ {𝑥𝐴𝜓}) = ({𝑥 ∣ (𝑥𝐴𝜑)} ∖ {𝑥 ∣ (𝑥𝐴𝜓)})
4 df-rab 2916 . . 3 {𝑥𝐴 ∣ (𝜑 ∧ ¬ 𝜓)} = {𝑥 ∣ (𝑥𝐴 ∧ (𝜑 ∧ ¬ 𝜓))}
5 difab 3874 . . . 4 ({𝑥 ∣ (𝑥𝐴𝜑)} ∖ {𝑥 ∣ (𝑥𝐴𝜓)}) = {𝑥 ∣ ((𝑥𝐴𝜑) ∧ ¬ (𝑥𝐴𝜓))}
6 anass 680 . . . . . 6 (((𝑥𝐴𝜑) ∧ ¬ 𝜓) ↔ (𝑥𝐴 ∧ (𝜑 ∧ ¬ 𝜓)))
7 simpr 477 . . . . . . . . 9 ((𝑥𝐴𝜓) → 𝜓)
87con3i 150 . . . . . . . 8 𝜓 → ¬ (𝑥𝐴𝜓))
98anim2i 592 . . . . . . 7 (((𝑥𝐴𝜑) ∧ ¬ 𝜓) → ((𝑥𝐴𝜑) ∧ ¬ (𝑥𝐴𝜓)))
10 pm3.2 463 . . . . . . . . . 10 (𝑥𝐴 → (𝜓 → (𝑥𝐴𝜓)))
1110adantr 481 . . . . . . . . 9 ((𝑥𝐴𝜑) → (𝜓 → (𝑥𝐴𝜓)))
1211con3d 148 . . . . . . . 8 ((𝑥𝐴𝜑) → (¬ (𝑥𝐴𝜓) → ¬ 𝜓))
1312imdistani 725 . . . . . . 7 (((𝑥𝐴𝜑) ∧ ¬ (𝑥𝐴𝜓)) → ((𝑥𝐴𝜑) ∧ ¬ 𝜓))
149, 13impbii 199 . . . . . 6 (((𝑥𝐴𝜑) ∧ ¬ 𝜓) ↔ ((𝑥𝐴𝜑) ∧ ¬ (𝑥𝐴𝜓)))
156, 14bitr3i 266 . . . . 5 ((𝑥𝐴 ∧ (𝜑 ∧ ¬ 𝜓)) ↔ ((𝑥𝐴𝜑) ∧ ¬ (𝑥𝐴𝜓)))
1615abbii 2736 . . . 4 {𝑥 ∣ (𝑥𝐴 ∧ (𝜑 ∧ ¬ 𝜓))} = {𝑥 ∣ ((𝑥𝐴𝜑) ∧ ¬ (𝑥𝐴𝜓))}
175, 16eqtr4i 2646 . . 3 ({𝑥 ∣ (𝑥𝐴𝜑)} ∖ {𝑥 ∣ (𝑥𝐴𝜓)}) = {𝑥 ∣ (𝑥𝐴 ∧ (𝜑 ∧ ¬ 𝜓))}
184, 17eqtr4i 2646 . 2 {𝑥𝐴 ∣ (𝜑 ∧ ¬ 𝜓)} = ({𝑥 ∣ (𝑥𝐴𝜑)} ∖ {𝑥 ∣ (𝑥𝐴𝜓)})
193, 18eqtr4i 2646 1 ({𝑥𝐴𝜑} ∖ {𝑥𝐴𝜓}) = {𝑥𝐴 ∣ (𝜑 ∧ ¬ 𝜓)}
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 384   = wceq 1480  wcel 1987  {cab 2607  {crab 2911  cdif 3553
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rab 2916  df-v 3188  df-dif 3559
This theorem is referenced by:  alephsuc3  9349  shftmbl  23219  musum  24824  clwwlknclwwlkdifs  26747  aciunf1  29317  poimirlem26  33088  poimirlem27  33089
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