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Theorem sbievOLD 2325
Description: Obsolete proof of sbiev 2324 as of 18-Jul-2023. (Contributed by Wolf Lammen, 18-Jan-2023.) (New usage is discouraged.) (Proof modification is discouraged.)
Hypotheses
Ref Expression
sbiev.1 𝑥𝜓
sbiev.2 (𝑥 = 𝑦 → (𝜑𝜓))
Assertion
Ref Expression
sbievOLD ([𝑦 / 𝑥]𝜑𝜓)
Distinct variable group:   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑥,𝑦)

Proof of Theorem sbievOLD
StepHypRef Expression
1 equsb1v 2110 . . 3 [𝑦 / 𝑥]𝑥 = 𝑦
2 sbiev.2 . . . 4 (𝑥 = 𝑦 → (𝜑𝜓))
32sbimi 2079 . . 3 ([𝑦 / 𝑥]𝑥 = 𝑦 → [𝑦 / 𝑥](𝜑𝜓))
41, 3ax-mp 5 . 2 [𝑦 / 𝑥](𝜑𝜓)
5 sbiev.1 . . . 4 𝑥𝜓
65sbf 2270 . . 3 ([𝑦 / 𝑥]𝜓𝜓)
76sblbisvOLD 2323 . 2 ([𝑦 / 𝑥](𝜑𝜓) ↔ ([𝑦 / 𝑥]𝜑𝜓))
84, 7mpbi 233 1 ([𝑦 / 𝑥]𝜑𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wnf 1785  [wsb 2069
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-10 2143  ax-12 2176
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-ex 1782  df-nf 1786  df-sb 2070
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator