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Theorem injust 2923
Description: Soundness justification theorem for df-in 2924. (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 2100 . . . 4 (𝑥 = 𝑧 → (𝑥𝐴𝑧𝐴))
2 eleq1 2100 . . . 4 (𝑥 = 𝑧 → (𝑥𝐵𝑧𝐵))
31, 2anbi12d 442 . . 3 (𝑥 = 𝑧 → ((𝑥𝐴𝑥𝐵) ↔ (𝑧𝐴𝑧𝐵)))
43cbvabv 2161 . 2 {𝑥 ∣ (𝑥𝐴𝑥𝐵)} = {𝑧 ∣ (𝑧𝐴𝑧𝐵)}
5 eleq1 2100 . . . 4 (𝑧 = 𝑦 → (𝑧𝐴𝑦𝐴))
6 eleq1 2100 . . . 4 (𝑧 = 𝑦 → (𝑧𝐵𝑦𝐵))
75, 6anbi12d 442 . . 3 (𝑧 = 𝑦 → ((𝑧𝐴𝑧𝐵) ↔ (𝑦𝐴𝑦𝐵)))
87cbvabv 2161 . 2 {𝑧 ∣ (𝑧𝐴𝑧𝐵)} = {𝑦 ∣ (𝑦𝐴𝑦𝐵)}
94, 8eqtri 2060 1 {𝑥 ∣ (𝑥𝐴𝑥𝐵)} = {𝑦 ∣ (𝑦𝐴𝑦𝐵)}
Colors of variables: wff set class
Syntax hints:  wa 97   = wceq 1243  wcel 1393  {cab 2026
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-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022
This theorem depends on definitions:  df-bi 110  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036
This theorem is referenced by: (None)
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