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Theorem difjust 3733
Description: Soundness justification theorem for df-dif 3734. (Contributed by Rodolfo Medina, 27-Apr-2010.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
Assertion
Ref Expression
difjust {𝑥 ∣ (𝑥𝐴 ∧ ¬ 𝑥𝐵)} = {𝑦 ∣ (𝑦𝐴 ∧ ¬ 𝑦𝐵)}
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑦,𝐴   𝑦,𝐵

Proof of Theorem difjust
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 eleq1w 2826 . . . 4 (𝑥 = 𝑧 → (𝑥𝐴𝑧𝐴))
2 eleq1w 2826 . . . . 5 (𝑥 = 𝑧 → (𝑥𝐵𝑧𝐵))
32notbid 309 . . . 4 (𝑥 = 𝑧 → (¬ 𝑥𝐵 ↔ ¬ 𝑧𝐵))
41, 3anbi12d 624 . . 3 (𝑥 = 𝑧 → ((𝑥𝐴 ∧ ¬ 𝑥𝐵) ↔ (𝑧𝐴 ∧ ¬ 𝑧𝐵)))
54cbvabv 2889 . 2 {𝑥 ∣ (𝑥𝐴 ∧ ¬ 𝑥𝐵)} = {𝑧 ∣ (𝑧𝐴 ∧ ¬ 𝑧𝐵)}
6 eleq1w 2826 . . . 4 (𝑧 = 𝑦 → (𝑧𝐴𝑦𝐴))
7 eleq1w 2826 . . . . 5 (𝑧 = 𝑦 → (𝑧𝐵𝑦𝐵))
87notbid 309 . . . 4 (𝑧 = 𝑦 → (¬ 𝑧𝐵 ↔ ¬ 𝑦𝐵))
96, 8anbi12d 624 . . 3 (𝑧 = 𝑦 → ((𝑧𝐴 ∧ ¬ 𝑧𝐵) ↔ (𝑦𝐴 ∧ ¬ 𝑦𝐵)))
109cbvabv 2889 . 2 {𝑧 ∣ (𝑧𝐴 ∧ ¬ 𝑧𝐵)} = {𝑦 ∣ (𝑦𝐴 ∧ ¬ 𝑦𝐵)}
115, 10eqtri 2786 1 {𝑥 ∣ (𝑥𝐴 ∧ ¬ 𝑥𝐵)} = {𝑦 ∣ (𝑦𝐴 ∧ ¬ 𝑦𝐵)}
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wa 384   = wceq 1652  wcel 2155  {cab 2750
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2069  ax-7 2105  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2349  ax-ext 2742
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2062  df-clab 2751  df-cleq 2757  df-clel 2760
This theorem is referenced by: (None)
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