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Theorem difjust 2913
Description: Soundness justification theorem for df-dif 2914. (Contributed by Rodolfo Medina, 27-Apr-2010.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
Assertion
Ref Expression
difjust {x ∣ (x A ¬ x B)} = {y ∣ (y A ¬ y B)}
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 eleq1 2097 . . . 4 (x = z → (x Az A))
2 eleq1 2097 . . . . 5 (x = z → (x Bz B))
32notbid 591 . . . 4 (x = z → (¬ x B ↔ ¬ z B))
41, 3anbi12d 442 . . 3 (x = z → ((x A ¬ x B) ↔ (z A ¬ z B)))
54cbvabv 2158 . 2 {x ∣ (x A ¬ x B)} = {z ∣ (z A ¬ z B)}
6 eleq1 2097 . . . 4 (z = y → (z Ay A))
7 eleq1 2097 . . . . 5 (z = y → (z By B))
87notbid 591 . . . 4 (z = y → (¬ z B ↔ ¬ y B))
96, 8anbi12d 442 . . 3 (z = y → ((z A ¬ z B) ↔ (y A ¬ y B)))
109cbvabv 2158 . 2 {z ∣ (z A ¬ z B)} = {y ∣ (y A ¬ y B)}
115, 10eqtri 2057 1 {x ∣ (x A ¬ x B)} = {y ∣ (y A ¬ y B)}
Colors of variables: wff set class
Syntax hints:  ¬ wn 3   wa 97   = wceq 1242   wcel 1390  {cab 2023
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-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033
This theorem is referenced by: (None)
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