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Theorem sb5rf 1729
 Description: Reversed substitution. (Contributed by NM, 3-Feb-2005.) (Proof shortened by Andrew Salmon, 25-May-2011.)
Hypothesis
Ref Expression
sb5rf.1 (φyφ)
Assertion
Ref Expression
sb5rf (φy(y = x [y / x]φ))

Proof of Theorem sb5rf
StepHypRef Expression
1 sb5rf.1 . . . 4 (φyφ)
21sbid2h 1726 . . 3 ([x / y][y / x]φφ)
3 sb1 1646 . . 3 ([x / y][y / x]φy(y = x [y / x]φ))
42, 3sylbir 125 . 2 (φy(y = x [y / x]φ))
5 stdpc7 1650 . . . 4 (y = x → ([y / x]φφ))
65imp 115 . . 3 ((y = x [y / x]φ) → φ)
71, 6exlimih 1481 . 2 (y(y = x [y / x]φ) → φ)
84, 7impbii 117 1 (φy(y = x [y / x]φ))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98  ∀wal 1240  ∃wex 1378  [wsb 1642 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1333  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-11 1394  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424 This theorem depends on definitions:  df-bi 110  df-sb 1643 This theorem is referenced by:  2sb5rf  1862  sbelx  1870
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