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Theorem sb5rf 1748
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 1745 . . 3 ([𝑥 / 𝑦][𝑦 / 𝑥]𝜑𝜑)
3 sb1 1665 . . 3 ([𝑥 / 𝑦][𝑦 / 𝑥]𝜑 → ∃𝑦(𝑦 = 𝑥 ∧ [𝑦 / 𝑥]𝜑))
42, 3sylbir 129 . 2 (𝜑 → ∃𝑦(𝑦 = 𝑥 ∧ [𝑦 / 𝑥]𝜑))
5 stdpc7 1669 . . . 4 (𝑦 = 𝑥 → ([𝑦 / 𝑥]𝜑𝜑))
65imp 119 . . 3 ((𝑦 = 𝑥 ∧ [𝑦 / 𝑥]𝜑) → 𝜑)
71, 6exlimih 1500 . 2 (∃𝑦(𝑦 = 𝑥 ∧ [𝑦 / 𝑥]𝜑) → 𝜑)
84, 7impbii 121 1 (𝜑 ↔ ∃𝑦(𝑦 = 𝑥 ∧ [𝑦 / 𝑥]𝜑))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101  wb 102  wal 1257  wex 1397  [wsb 1661
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-5 1352  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-11 1413  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443
This theorem depends on definitions:  df-bi 114  df-sb 1662
This theorem is referenced by:  2sb5rf  1881  sbelx  1889
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