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Theorem sb5rf 2498
Description: Reversed substitution. (Contributed by NM, 3-Feb-2005.) (Revised by Mario Carneiro, 6-Oct-2016.) (Proof shortened by Wolf Lammen, 20-Sep-2018.)
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 2229 . . 3 (𝑦 = 𝑥 → ([𝑦 / 𝑥]𝜑𝜑))
31, 2equsex 2382 . 2 (∃𝑦(𝑦 = 𝑥 ∧ [𝑦 / 𝑥]𝜑) ↔ 𝜑)
43bicomi 216 1 (𝜑 ↔ ∃𝑦(𝑦 = 𝑥 ∧ [𝑦 / 𝑥]𝜑))
Colors of variables: wff setvar class
Syntax hints:  wb 198  wa 386  wex 1823  wnf 1827  [wsb 2011
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2054  ax-12 2162  ax-13 2333
This theorem depends on definitions:  df-bi 199  df-an 387  df-ex 1824  df-nf 1828  df-sb 2012
This theorem is referenced by: (None)
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