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Theorem sb9i 2094
 Description: Commutation of quantification and substitution variables. (Contributed by NM, 5-Aug-1993.)
Assertion
Ref Expression
sb9i (x[x / y]φy[y / x]φ)

Proof of Theorem sb9i
StepHypRef Expression
1 drsb1 2022 . . . . 5 (y y = x → ([y / y]φ ↔ [y / x]φ))
2 drsb2 2061 . . . . 5 (y y = x → ([y / y]φ ↔ [x / y]φ))
31, 2bitr3d 246 . . . 4 (y y = x → ([y / x]φ ↔ [x / y]φ))
43dral1 1965 . . 3 (y y = x → (y[y / x]φx[x / y]φ))
54biimprd 214 . 2 (y y = x → (x[x / y]φy[y / x]φ))
6 nfnae 1956 . . . 4 x ¬ y y = x
7 hbsb2 2057 . . . 4 y y = x → ([x / y]φy[x / y]φ))
86, 7alimd 1764 . . 3 y y = x → (x[x / y]φxy[x / y]φ))
9 stdpc4 2024 . . . . . 6 (x[x / y]φ → [y / x][x / y]φ)
10 sbco 2083 . . . . . 6 ([y / x][x / y]φ ↔ [y / x]φ)
119, 10sylib 188 . . . . 5 (x[x / y]φ → [y / x]φ)
1211alimi 1559 . . . 4 (yx[x / y]φy[y / x]φ)
1312a7s 1735 . . 3 (xy[x / y]φy[y / x]φ)
148, 13syl6 29 . 2 y y = x → (x[x / y]φy[y / x]φ))
155, 14pm2.61i 156 1 (x[x / y]φy[y / x]φ)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1540  [wsb 1648 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 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-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649 This theorem is referenced by:  sb9  2095
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