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Theorem difrab 3205
Description: Difference of two restricted class abstractions. (Contributed by NM, 23-Oct-2004.)
Assertion
Ref Expression
difrab ({x Aφ} ∖ {x Aψ}) = {x A ∣ (φ ¬ ψ)}

Proof of Theorem difrab
StepHypRef Expression
1 df-rab 2309 . . 3 {x Aφ} = {x ∣ (x A φ)}
2 df-rab 2309 . . 3 {x Aψ} = {x ∣ (x A ψ)}
31, 2difeq12i 3054 . 2 ({x Aφ} ∖ {x Aψ}) = ({x ∣ (x A φ)} ∖ {x ∣ (x A ψ)})
4 df-rab 2309 . . 3 {x A ∣ (φ ¬ ψ)} = {x ∣ (x A (φ ¬ ψ))}
5 difab 3200 . . . 4 ({x ∣ (x A φ)} ∖ {x ∣ (x A ψ)}) = {x ∣ ((x A φ) ¬ (x A ψ))}
6 anass 381 . . . . . 6 (((x A φ) ¬ ψ) ↔ (x A (φ ¬ ψ)))
7 simpr 103 . . . . . . . . 9 ((x A ψ) → ψ)
87con3i 561 . . . . . . . 8 ψ → ¬ (x A ψ))
98anim2i 324 . . . . . . 7 (((x A φ) ¬ ψ) → ((x A φ) ¬ (x A ψ)))
10 pm3.2 126 . . . . . . . . . 10 (x A → (ψ → (x A ψ)))
1110adantr 261 . . . . . . . . 9 ((x A φ) → (ψ → (x A ψ)))
1211con3d 560 . . . . . . . 8 ((x A φ) → (¬ (x A ψ) → ¬ ψ))
1312imdistani 419 . . . . . . 7 (((x A φ) ¬ (x A ψ)) → ((x A φ) ¬ ψ))
149, 13impbii 117 . . . . . 6 (((x A φ) ¬ ψ) ↔ ((x A φ) ¬ (x A ψ)))
156, 14bitr3i 175 . . . . 5 ((x A (φ ¬ ψ)) ↔ ((x A φ) ¬ (x A ψ)))
1615abbii 2150 . . . 4 {x ∣ (x A (φ ¬ ψ))} = {x ∣ ((x A φ) ¬ (x A ψ))}
175, 16eqtr4i 2060 . . 3 ({x ∣ (x A φ)} ∖ {x ∣ (x A ψ)}) = {x ∣ (x A (φ ¬ ψ))}
184, 17eqtr4i 2060 . 2 {x A ∣ (φ ¬ ψ)} = ({x ∣ (x A φ)} ∖ {x ∣ (x A ψ)})
193, 18eqtr4i 2060 1 ({x Aφ} ∖ {x Aψ}) = {x A ∣ (φ ¬ ψ)}
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4   wa 97   = wceq 1242   wcel 1390  {cab 2023  {crab 2304  cdif 2908
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-tru 1245  df-fal 1248  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rab 2309  df-v 2553  df-dif 2914
This theorem is referenced by: (None)
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