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Theorem sb5rf 1863
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 1860 . . 3 ([𝑥 / 𝑦][𝑦 / 𝑥]𝜑𝜑)
3 sb1 1777 . . 3 ([𝑥 / 𝑦][𝑦 / 𝑥]𝜑 → ∃𝑦(𝑦 = 𝑥 ∧ [𝑦 / 𝑥]𝜑))
42, 3sylbir 135 . 2 (𝜑 → ∃𝑦(𝑦 = 𝑥 ∧ [𝑦 / 𝑥]𝜑))
5 stdpc7 1781 . . . 4 (𝑦 = 𝑥 → ([𝑦 / 𝑥]𝜑𝜑))
65imp 124 . . 3 ((𝑦 = 𝑥 ∧ [𝑦 / 𝑥]𝜑) → 𝜑)
71, 6exlimih 1604 . 2 (∃𝑦(𝑦 = 𝑥 ∧ [𝑦 / 𝑥]𝜑) → 𝜑)
84, 7impbii 126 1 (𝜑 ↔ ∃𝑦(𝑦 = 𝑥 ∧ [𝑦 / 𝑥]𝜑))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wb 105  wal 1362  wex 1503  [wsb 1773
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1458  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-11 1517  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545
This theorem depends on definitions:  df-bi 117  df-sb 1774
This theorem is referenced by:  2sb5rf  2001  sbelx  2009
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