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Theorem sb5rf 1824
 Description: Reversed substitution. (Contributed by NM, 3-Feb-2005.) (Proof shortened by Andrew Salmon, 25-May-2011.)
Hypothesis
Ref Expression
sb5rf.1 (𝜑 → ∀𝑦𝜑)
Assertion
Ref Expression
sb5rf (𝜑 ↔ ∃𝑦(𝑦 = 𝑥 ∧ [𝑦 / 𝑥]𝜑))

Proof of Theorem sb5rf
StepHypRef Expression
1 sb5rf.1 . . . 4 (𝜑 → ∀𝑦𝜑)
21sbid2h 1821 . . 3 ([𝑥 / 𝑦][𝑦 / 𝑥]𝜑𝜑)
3 sb1 1739 . . 3 ([𝑥 / 𝑦][𝑦 / 𝑥]𝜑 → ∃𝑦(𝑦 = 𝑥 ∧ [𝑦 / 𝑥]𝜑))
42, 3sylbir 134 . 2 (𝜑 → ∃𝑦(𝑦 = 𝑥 ∧ [𝑦 / 𝑥]𝜑))
5 stdpc7 1743 . . . 4 (𝑦 = 𝑥 → ([𝑦 / 𝑥]𝜑𝜑))
65imp 123 . . 3 ((𝑦 = 𝑥 ∧ [𝑦 / 𝑥]𝜑) → 𝜑)
71, 6exlimih 1572 . 2 (∃𝑦(𝑦 = 𝑥 ∧ [𝑦 / 𝑥]𝜑) → 𝜑)
84, 7impbii 125 1 (𝜑 ↔ ∃𝑦(𝑦 = 𝑥 ∧ [𝑦 / 𝑥]𝜑))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104  ∀wal 1329  ∃wex 1468  [wsb 1735 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1423  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-11 1484  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514 This theorem depends on definitions:  df-bi 116  df-sb 1736 This theorem is referenced by:  2sb5rf  1964  sbelx  1972
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