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Theorem sb5OLD 2273
Description: Obsolete version of sb5 2272 as of 21-Sep-2024. (Contributed by NM, 18-Aug-1993.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
sb5OLD ([𝑦 / 𝑥]𝜑 ↔ ∃𝑥(𝑥 = 𝑦𝜑))
Distinct variable group:   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem sb5OLD
StepHypRef Expression
1 nfs1v 2157 . . 3 𝑥[𝑦 / 𝑥]𝜑
2 sbequ12 2248 . . 3 (𝑥 = 𝑦 → (𝜑 ↔ [𝑦 / 𝑥]𝜑))
31, 2equsexv 2264 . 2 (∃𝑥(𝑥 = 𝑦𝜑) ↔ [𝑦 / 𝑥]𝜑)
43bicomi 223 1 ([𝑦 / 𝑥]𝜑 ↔ ∃𝑥(𝑥 = 𝑦𝜑))
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 396  wex 1786  [wsb 2071
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2015  ax-10 2141  ax-12 2175
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-ex 1787  df-nf 1791  df-sb 2072
This theorem is referenced by: (None)
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