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Theorem cbvabwOLD 2871
Description: Obsolete version of cbvabw 2870 as of 23-May-2024. (Contributed by Andrew Salmon, 11-Jul-2011.) (Revised by Gino Giotto, 10-Jan-2024.) (Proof modification is discouraged.) (New usage is discouraged.)
Hypotheses
Ref Expression
cbvabwOLD.1 𝑦𝜑
cbvabwOLD.2 𝑥𝜓
cbvabwOLD.3 (𝑥 = 𝑦 → (𝜑𝜓))
Assertion
Ref Expression
cbvabwOLD {𝑥𝜑} = {𝑦𝜓}
Distinct variable group:   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑥,𝑦)

Proof of Theorem cbvabwOLD
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 cbvabwOLD.1 . . . . 5 𝑦𝜑
21sbco2v 2344 . . . 4 ([𝑧 / 𝑦][𝑦 / 𝑥]𝜑 ↔ [𝑧 / 𝑥]𝜑)
3 cbvabwOLD.2 . . . . . 6 𝑥𝜓
4 cbvabwOLD.3 . . . . . 6 (𝑥 = 𝑦 → (𝜑𝜓))
53, 4sbiev 2324 . . . . 5 ([𝑦 / 𝑥]𝜑𝜓)
65sbbii 2081 . . . 4 ([𝑧 / 𝑦][𝑦 / 𝑥]𝜑 ↔ [𝑧 / 𝑦]𝜓)
72, 6bitr3i 280 . . 3 ([𝑧 / 𝑥]𝜑 ↔ [𝑧 / 𝑦]𝜓)
8 df-clab 2780 . . 3 (𝑧 ∈ {𝑥𝜑} ↔ [𝑧 / 𝑥]𝜑)
9 df-clab 2780 . . 3 (𝑧 ∈ {𝑦𝜓} ↔ [𝑧 / 𝑦]𝜓)
107, 8, 93bitr4i 306 . 2 (𝑧 ∈ {𝑥𝜑} ↔ 𝑧 ∈ {𝑦𝜓})
1110eqriv 2798 1 {𝑥𝜑} = {𝑦𝜓}
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209   = wceq 1538  wnf 1785  [wsb 2069  wcel 2112  {cab 2779
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773
This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2780  df-cleq 2794
This theorem is referenced by: (None)
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