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Theorem sbco 2083
 Description: A composition law for substitution. (Contributed by NM, 5-Aug-1993.)
Assertion
Ref Expression
sbco ([y / x][x / y]φ ↔ [y / x]φ)

Proof of Theorem sbco
StepHypRef Expression
1 equsb2 2035 . . 3 [y / x]y = x
2 sbequ12 1919 . . . . 5 (y = x → (φ ↔ [x / y]φ))
32bicomd 192 . . . 4 (y = x → ([x / y]φφ))
43sbimi 1652 . . 3 ([y / x]y = x → [y / x]([x / y]φφ))
51, 4ax-mp 8 . 2 [y / x]([x / y]φφ)
6 sbbi 2071 . 2 ([y / x]([x / y]φφ) ↔ ([y / x][x / y]φ ↔ [y / x]φ))
75, 6mpbi 199 1 ([y / x][x / y]φ ↔ [y / x]φ)
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 176  [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:  sbid2  2084  sbco3  2088  sb9i  2094
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