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Theorem injust 3046
Description: Soundness justification theorem for df-in 3047. (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 2180 . . . 4 (𝑥 = 𝑧 → (𝑥𝐴𝑧𝐴))
2 eleq1 2180 . . . 4 (𝑥 = 𝑧 → (𝑥𝐵𝑧𝐵))
31, 2anbi12d 464 . . 3 (𝑥 = 𝑧 → ((𝑥𝐴𝑥𝐵) ↔ (𝑧𝐴𝑧𝐵)))
43cbvabv 2241 . 2 {𝑥 ∣ (𝑥𝐴𝑥𝐵)} = {𝑧 ∣ (𝑧𝐴𝑧𝐵)}
5 eleq1 2180 . . . 4 (𝑧 = 𝑦 → (𝑧𝐴𝑦𝐴))
6 eleq1 2180 . . . 4 (𝑧 = 𝑦 → (𝑧𝐵𝑦𝐵))
75, 6anbi12d 464 . . 3 (𝑧 = 𝑦 → ((𝑧𝐴𝑧𝐵) ↔ (𝑦𝐴𝑦𝐵)))
87cbvabv 2241 . 2 {𝑧 ∣ (𝑧𝐴𝑧𝐵)} = {𝑦 ∣ (𝑦𝐴𝑦𝐵)}
94, 8eqtri 2138 1 {𝑥 ∣ (𝑥𝐴𝑥𝐵)} = {𝑦 ∣ (𝑦𝐴𝑦𝐵)}
Colors of variables: wff set class
Syntax hints:  wa 103   = wceq 1316  wcel 1465  {cab 2103
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113
This theorem is referenced by: (None)
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