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Theorem rabnc 3574
 Description: Law of noncontradiction, in terms of restricted class abstractions. (Contributed by Jeff Madsen, 20-Jun-2011.)
Assertion
Ref Expression
rabnc ({x A φ} ∩ {x A ¬ φ}) =
Distinct variable group:   x,A
Allowed substitution hint:   φ(x)

Proof of Theorem rabnc
StepHypRef Expression
1 inrab 3527 . 2 ({x A φ} ∩ {x A ¬ φ}) = {x A (φ ¬ φ)}
2 rabeq0 3572 . . 3 ({x A (φ ¬ φ)} = x A ¬ (φ ¬ φ))
3 pm3.24 852 . . . 4 ¬ (φ ¬ φ)
43a1i 10 . . 3 (x A → ¬ (φ ¬ φ))
52, 4mprgbir 2684 . 2 {x A (φ ¬ φ)} =
61, 5eqtri 2373 1 ({x A φ} ∩ {x A ¬ φ}) =
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   ∧ wa 358   = wceq 1642   ∈ wcel 1710  {crab 2618   ∩ cin 3208  ∅c0 3550 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ne 2518  df-ral 2619  df-rex 2620  df-rab 2623  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-dif 3215  df-nul 3551 This theorem is referenced by: (None)
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