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Theorem sbidm 1728
Description: An idempotent law for substitution. (Contributed by NM, 30-Jun-1994.) (Proof rewritten by Jim Kingdon, 21-Jan-2018.)
Assertion
Ref Expression
sbidm ([y / x][y / x]φ ↔ [y / x]φ)

Proof of Theorem sbidm
StepHypRef Expression
1 df-sb 1643 . . . . 5 ([y / x]φ ↔ ((x = yφ) x(x = y φ)))
21simplbi 259 . . . 4 ([y / x]φ → (x = yφ))
32sbimi 1644 . . 3 ([y / x][y / x]φ → [y / x](x = yφ))
4 sbequ8 1724 . . 3 ([y / x]φ ↔ [y / x](x = yφ))
53, 4sylibr 137 . 2 ([y / x][y / x]φ → [y / x]φ)
6 ax-1 5 . . 3 ([y / x]φ → (x = y → [y / x]φ))
7 sb1 1646 . . . 4 ([y / x]φx(x = y φ))
8 pm4.24 375 . . . . . . . 8 (x(x = y φ) ↔ (x(x = y φ) x(x = y φ)))
9 ax-ie1 1379 . . . . . . . . 9 (x(x = y φ) → xx(x = y φ))
10919.41h 1572 . . . . . . . 8 (x((x = y φ) x(x = y φ)) ↔ (x(x = y φ) x(x = y φ)))
118, 10bitr4i 176 . . . . . . 7 (x(x = y φ) ↔ x((x = y φ) x(x = y φ)))
12 ax-1 5 . . . . . . . . . 10 (φ → (x = yφ))
1312anim2i 324 . . . . . . . . 9 ((x = y φ) → (x = y (x = yφ)))
1413anim1i 323 . . . . . . . 8 (((x = y φ) x(x = y φ)) → ((x = y (x = yφ)) x(x = y φ)))
1514eximi 1488 . . . . . . 7 (x((x = y φ) x(x = y φ)) → x((x = y (x = yφ)) x(x = y φ)))
1611, 15sylbi 114 . . . . . 6 (x(x = y φ) → x((x = y (x = yφ)) x(x = y φ)))
17 anass 381 . . . . . . 7 (((x = y (x = yφ)) x(x = y φ)) ↔ (x = y ((x = yφ) x(x = y φ))))
1817exbii 1493 . . . . . 6 (x((x = y (x = yφ)) x(x = y φ)) ↔ x(x = y ((x = yφ) x(x = y φ))))
1916, 18sylib 127 . . . . 5 (x(x = y φ) → x(x = y ((x = yφ) x(x = y φ))))
201anbi2i 430 . . . . . 6 ((x = y [y / x]φ) ↔ (x = y ((x = yφ) x(x = y φ))))
2120exbii 1493 . . . . 5 (x(x = y [y / x]φ) ↔ x(x = y ((x = yφ) x(x = y φ))))
2219, 21sylibr 137 . . . 4 (x(x = y φ) → x(x = y [y / x]φ))
237, 22syl 14 . . 3 ([y / x]φx(x = y [y / x]φ))
24 df-sb 1643 . . 3 ([y / x][y / x]φ ↔ ((x = y → [y / x]φ) x(x = y [y / x]φ)))
256, 23, 24sylanbrc 394 . 2 ([y / x]φ → [y / x][y / x]φ)
265, 25impbii 117 1 ([y / x][y / x]φ ↔ [y / x]φ)
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98  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-4 1397  ax-ial 1424
This theorem depends on definitions:  df-bi 110  df-sb 1643
This theorem is referenced by: (None)
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