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Theorem injust 2987
Description: Soundness justification theorem for df-in 2988. (Contributed by Rodolfo Medina, 28-Apr-2010.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
Assertion
Ref Expression
injust {𝑥 ∣ (𝑥𝐴𝑥𝐵)} = {𝑦 ∣ (𝑦𝐴𝑦𝐵)}
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑦,𝐴   𝑦,𝐵

Proof of Theorem injust
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 eleq1 2145 . . . 4 (𝑥 = 𝑧 → (𝑥𝐴𝑧𝐴))
2 eleq1 2145 . . . 4 (𝑥 = 𝑧 → (𝑥𝐵𝑧𝐵))
31, 2anbi12d 457 . . 3 (𝑥 = 𝑧 → ((𝑥𝐴𝑥𝐵) ↔ (𝑧𝐴𝑧𝐵)))
43cbvabv 2206 . 2 {𝑥 ∣ (𝑥𝐴𝑥𝐵)} = {𝑧 ∣ (𝑧𝐴𝑧𝐵)}
5 eleq1 2145 . . . 4 (𝑧 = 𝑦 → (𝑧𝐴𝑦𝐴))
6 eleq1 2145 . . . 4 (𝑧 = 𝑦 → (𝑧𝐵𝑦𝐵))
75, 6anbi12d 457 . . 3 (𝑧 = 𝑦 → ((𝑧𝐴𝑧𝐵) ↔ (𝑦𝐴𝑦𝐵)))
87cbvabv 2206 . 2 {𝑧 ∣ (𝑧𝐴𝑧𝐵)} = {𝑦 ∣ (𝑦𝐴𝑦𝐵)}
94, 8eqtri 2103 1 {𝑥 ∣ (𝑥𝐴𝑥𝐵)} = {𝑦 ∣ (𝑦𝐴𝑦𝐵)}
Colors of variables: wff set class
Syntax hints:  wa 102   = wceq 1285  wcel 1434  {cab 2069
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065
This theorem depends on definitions:  df-bi 115  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079
This theorem is referenced by: (None)
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