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Theorem bj-sbidmOLD 34061
 Description: Obsolete proof of sbidm 2550 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 2529 . . 3 [𝑦 / 𝑥]𝑦 = 𝑥
2 sbequ12r 2247 . . . 4 (𝑦 = 𝑥 → ([𝑦 / 𝑥]𝜑𝜑))
32sbimi 2072 . . 3 ([𝑦 / 𝑥]𝑦 = 𝑥 → [𝑦 / 𝑥]([𝑦 / 𝑥]𝜑𝜑))
41, 3ax-mp 5 . 2 [𝑦 / 𝑥]([𝑦 / 𝑥]𝜑𝜑)
5 sbbi 2311 . 2 ([𝑦 / 𝑥]([𝑦 / 𝑥]𝜑𝜑) ↔ ([𝑦 / 𝑥][𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑))
64, 5mpbi 231 1 ([𝑦 / 𝑥][𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑)
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 207  [wsb 2062 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-10 2138  ax-12 2169  ax-13 2385 This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-ex 1774  df-nf 1778  df-sb 2063 This theorem is referenced by: (None)
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