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Theorem difrab 3437
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 2484 . . 3 {𝑥𝐴𝜑} = {𝑥 ∣ (𝑥𝐴𝜑)}
2 df-rab 2484 . . 3 {𝑥𝐴𝜓} = {𝑥 ∣ (𝑥𝐴𝜓)}
31, 2difeq12i 3279 . 2 ({𝑥𝐴𝜑} ∖ {𝑥𝐴𝜓}) = ({𝑥 ∣ (𝑥𝐴𝜑)} ∖ {𝑥 ∣ (𝑥𝐴𝜓)})
4 df-rab 2484 . . 3 {𝑥𝐴 ∣ (𝜑 ∧ ¬ 𝜓)} = {𝑥 ∣ (𝑥𝐴 ∧ (𝜑 ∧ ¬ 𝜓))}
5 difab 3432 . . . 4 ({𝑥 ∣ (𝑥𝐴𝜑)} ∖ {𝑥 ∣ (𝑥𝐴𝜓)}) = {𝑥 ∣ ((𝑥𝐴𝜑) ∧ ¬ (𝑥𝐴𝜓))}
6 anass 401 . . . . . 6 (((𝑥𝐴𝜑) ∧ ¬ 𝜓) ↔ (𝑥𝐴 ∧ (𝜑 ∧ ¬ 𝜓)))
7 simpr 110 . . . . . . . . 9 ((𝑥𝐴𝜓) → 𝜓)
87con3i 633 . . . . . . . 8 𝜓 → ¬ (𝑥𝐴𝜓))
98anim2i 342 . . . . . . 7 (((𝑥𝐴𝜑) ∧ ¬ 𝜓) → ((𝑥𝐴𝜑) ∧ ¬ (𝑥𝐴𝜓)))
10 pm3.2 139 . . . . . . . . . 10 (𝑥𝐴 → (𝜓 → (𝑥𝐴𝜓)))
1110adantr 276 . . . . . . . . 9 ((𝑥𝐴𝜑) → (𝜓 → (𝑥𝐴𝜓)))
1211con3d 632 . . . . . . . 8 ((𝑥𝐴𝜑) → (¬ (𝑥𝐴𝜓) → ¬ 𝜓))
1312imdistani 445 . . . . . . 7 (((𝑥𝐴𝜑) ∧ ¬ (𝑥𝐴𝜓)) → ((𝑥𝐴𝜑) ∧ ¬ 𝜓))
149, 13impbii 126 . . . . . 6 (((𝑥𝐴𝜑) ∧ ¬ 𝜓) ↔ ((𝑥𝐴𝜑) ∧ ¬ (𝑥𝐴𝜓)))
156, 14bitr3i 186 . . . . 5 ((𝑥𝐴 ∧ (𝜑 ∧ ¬ 𝜓)) ↔ ((𝑥𝐴𝜑) ∧ ¬ (𝑥𝐴𝜓)))
1615abbii 2312 . . . 4 {𝑥 ∣ (𝑥𝐴 ∧ (𝜑 ∧ ¬ 𝜓))} = {𝑥 ∣ ((𝑥𝐴𝜑) ∧ ¬ (𝑥𝐴𝜓))}
175, 16eqtr4i 2220 . . 3 ({𝑥 ∣ (𝑥𝐴𝜑)} ∖ {𝑥 ∣ (𝑥𝐴𝜓)}) = {𝑥 ∣ (𝑥𝐴 ∧ (𝜑 ∧ ¬ 𝜓))}
184, 17eqtr4i 2220 . 2 {𝑥𝐴 ∣ (𝜑 ∧ ¬ 𝜓)} = ({𝑥 ∣ (𝑥𝐴𝜑)} ∖ {𝑥 ∣ (𝑥𝐴𝜓)})
193, 18eqtr4i 2220 1 ({𝑥𝐴𝜑} ∖ {𝑥𝐴𝜓}) = {𝑥𝐴 ∣ (𝜑 ∧ ¬ 𝜓)}
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 104   = wceq 1364  wcel 2167  {cab 2182  {crab 2479  cdif 3154
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178
This theorem depends on definitions:  df-bi 117  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rab 2484  df-v 2765  df-dif 3159
This theorem is referenced by: (None)
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