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Theorem unjust 2948
Description: Soundness justification theorem for df-un 2949. (Contributed by Rodolfo Medina, 28-Apr-2010.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
Assertion
Ref Expression
unjust {𝑥 ∣ (𝑥𝐴𝑥𝐵)} = {𝑦 ∣ (𝑦𝐴𝑦𝐵)}
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑦,𝐴   𝑦,𝐵

Proof of Theorem unjust
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 eleq1 2116 . . . 4 (𝑥 = 𝑧 → (𝑥𝐴𝑧𝐴))
2 eleq1 2116 . . . 4 (𝑥 = 𝑧 → (𝑥𝐵𝑧𝐵))
31, 2orbi12d 717 . . 3 (𝑥 = 𝑧 → ((𝑥𝐴𝑥𝐵) ↔ (𝑧𝐴𝑧𝐵)))
43cbvabv 2177 . 2 {𝑥 ∣ (𝑥𝐴𝑥𝐵)} = {𝑧 ∣ (𝑧𝐴𝑧𝐵)}
5 eleq1 2116 . . . 4 (𝑧 = 𝑦 → (𝑧𝐴𝑦𝐴))
6 eleq1 2116 . . . 4 (𝑧 = 𝑦 → (𝑧𝐵𝑦𝐵))
75, 6orbi12d 717 . . 3 (𝑧 = 𝑦 → ((𝑧𝐴𝑧𝐵) ↔ (𝑦𝐴𝑦𝐵)))
87cbvabv 2177 . 2 {𝑧 ∣ (𝑧𝐴𝑧𝐵)} = {𝑦 ∣ (𝑦𝐴𝑦𝐵)}
94, 8eqtri 2076 1 {𝑥 ∣ (𝑥𝐴𝑥𝐵)} = {𝑦 ∣ (𝑦𝐴𝑦𝐵)}
Colors of variables: wff set class
Syntax hints:  wo 639   = wceq 1259  wcel 1409  {cab 2042
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052
This theorem is referenced by: (None)
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