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Theorem sbequ2OLDOLD 2516
Description: Obsolete version of sbequ2 2252 as of 8-Jul-2023. An equality theorem for substitution. (Contributed by NM, 16-May-1993.) (Proof shortened by Wolf Lammen, 25-Feb-2018.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
sbequ2OLDOLD (𝑥 = 𝑦 → ([𝑦 / 𝑥]𝜑𝜑))

Proof of Theorem sbequ2OLDOLD
StepHypRef Expression
1 dfsb1 2512 . . 3 ([𝑦 / 𝑥]𝜑 ↔ ((𝑥 = 𝑦𝜑) ∧ ∃𝑥(𝑥 = 𝑦𝜑)))
21simplbi 501 . 2 ([𝑦 / 𝑥]𝜑 → (𝑥 = 𝑦𝜑))
32com12 32 1 (𝑥 = 𝑦 → ([𝑦 / 𝑥]𝜑𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  wex 1781  [wsb 2070
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 1912  ax-6 1971  ax-7 2016  ax-10 2146  ax-12 2179  ax-13 2392
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-ex 1782  df-nf 1786  df-sb 2071
This theorem is referenced by: (None)
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