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Theorem bj-sbidmOLD 31827
Description: Obsolete proof temporarily kept here to check it gives no additional insight. (Contributed by NM, 8-Mar-1995.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
bj-sbidmOLD ([𝑦 / 𝑥][𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑)

Proof of Theorem bj-sbidmOLD
StepHypRef Expression
1 equsb2 2356 . . 3 [𝑦 / 𝑥]𝑦 = 𝑥
2 sbequ12r 2097 . . . 4 (𝑦 = 𝑥 → ([𝑦 / 𝑥]𝜑𝜑))
32sbimi 1872 . . 3 ([𝑦 / 𝑥]𝑦 = 𝑥 → [𝑦 / 𝑥]([𝑦 / 𝑥]𝜑𝜑))
41, 3ax-mp 5 . 2 [𝑦 / 𝑥]([𝑦 / 𝑥]𝜑𝜑)
5 sbbi 2388 . 2 ([𝑦 / 𝑥]([𝑦 / 𝑥]𝜑𝜑) ↔ ([𝑦 / 𝑥][𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑))
64, 5mpbi 218 1 ([𝑦 / 𝑥][𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑)
Colors of variables: wff setvar class
Syntax hints:  wb 194  [wsb 1866
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-10 2005  ax-12 2033  ax-13 2233
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867
This theorem is referenced by: (None)
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