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Theorem sb5rf 2469
Description: Reversed substitution. Usage of this theorem is discouraged because it depends on ax-13 2374. (Contributed by NM, 3-Feb-2005.) (Revised by Mario Carneiro, 6-Oct-2016.) (Proof shortened by Wolf Lammen, 20-Sep-2018.) (New usage is discouraged.)
Hypothesis
Ref Expression
sb5rf.1 𝑦𝜑
Assertion
Ref Expression
sb5rf (𝜑 ↔ ∃𝑦(𝑦 = 𝑥 ∧ [𝑦 / 𝑥]𝜑))

Proof of Theorem sb5rf
StepHypRef Expression
1 sb5rf.1 . . 3 𝑦𝜑
2 sbequ12r 2249 . . 3 (𝑦 = 𝑥 → ([𝑦 / 𝑥]𝜑𝜑))
31, 2equsex 2420 . 2 (∃𝑦(𝑦 = 𝑥 ∧ [𝑦 / 𝑥]𝜑) ↔ 𝜑)
43bicomi 223 1 (𝜑 ↔ ∃𝑦(𝑦 = 𝑥 ∧ [𝑦 / 𝑥]𝜑))
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 396  wex 1786  wnf 1790  [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-12 2175  ax-13 2374
This theorem depends on definitions:  df-bi 206  df-an 397  df-ex 1787  df-nf 1791  df-sb 2072
This theorem is referenced by: (None)
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