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Theorem sb5rf 2090
 Description: Reversed substitution. (Contributed by NM, 3-Feb-2005.) (Revised by Mario Carneiro, 6-Oct-2016.)
Hypothesis
Ref Expression
sb5rf.1 yφ
Assertion
Ref Expression
sb5rf (φy(y = x [y / x]φ))

Proof of Theorem sb5rf
StepHypRef Expression
1 sb5rf.1 . . . 4 yφ
21sbid2 2084 . . 3 ([x / y][y / x]φφ)
3 sb1 1651 . . 3 ([x / y][y / x]φy(y = x [y / x]φ))
42, 3sylbir 204 . 2 (φy(y = x [y / x]φ))
5 stdpc7 1917 . . . 4 (y = x → ([y / x]φφ))
65imp 418 . . 3 ((y = x [y / x]φ) → φ)
71, 6exlimi 1803 . 2 (y(y = x [y / x]φ) → φ)
84, 7impbii 180 1 (φy(y = x [y / x]φ))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 176   ∧ wa 358  ∃wex 1541  Ⅎwnf 1544  [wsb 1648 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925 This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649 This theorem is referenced by:  2sb5rf  2117
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