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Theorem unjust 3132
Description: Soundness justification theorem for df-un 3133. (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 2240 . . . 4 (𝑥 = 𝑧 → (𝑥𝐴𝑧𝐴))
2 eleq1 2240 . . . 4 (𝑥 = 𝑧 → (𝑥𝐵𝑧𝐵))
31, 2orbi12d 793 . . 3 (𝑥 = 𝑧 → ((𝑥𝐴𝑥𝐵) ↔ (𝑧𝐴𝑧𝐵)))
43cbvabv 2302 . 2 {𝑥 ∣ (𝑥𝐴𝑥𝐵)} = {𝑧 ∣ (𝑧𝐴𝑧𝐵)}
5 eleq1 2240 . . . 4 (𝑧 = 𝑦 → (𝑧𝐴𝑦𝐴))
6 eleq1 2240 . . . 4 (𝑧 = 𝑦 → (𝑧𝐵𝑦𝐵))
75, 6orbi12d 793 . . 3 (𝑧 = 𝑦 → ((𝑧𝐴𝑧𝐵) ↔ (𝑦𝐴𝑦𝐵)))
87cbvabv 2302 . 2 {𝑧 ∣ (𝑧𝐴𝑧𝐵)} = {𝑦 ∣ (𝑦𝐴𝑦𝐵)}
94, 8eqtri 2198 1 {𝑥 ∣ (𝑥𝐴𝑥𝐵)} = {𝑦 ∣ (𝑦𝐴𝑦𝐵)}
Colors of variables: wff set class
Syntax hints:  wo 708   = wceq 1353  wcel 2148  {cab 2163
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173
This theorem is referenced by: (None)
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