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Theorem sbco 1839
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 1666 . . 3 [y / x]y = x
2 sbequ12 1651 . . . . 5 (y = x → (φ ↔ [x / y]φ))
32bicomd 129 . . . 4 (y = x → ([x / y]φφ))
43sbimi 1644 . . 3 ([y / x]y = x → [y / x]([x / y]φφ))
51, 4ax-mp 7 . 2 [y / x]([x / y]φφ)
6 sbbi 1830 . 2 ([y / x]([x / y]φφ) ↔ ([y / x][x / y]φ ↔ [y / x]φ))
75, 6mpbi 133 1 ([y / x][x / y]φ ↔ [y / x]φ)
Colors of variables: wff set class
Syntax hints:  wb 98  [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-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425
This theorem depends on definitions:  df-bi 110  df-nf 1347  df-sb 1643
This theorem is referenced by:  sbco3v  1840
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