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Theorem sbco3 2088
 Description: A composition law for substitution. (Contributed by NM, 5-Aug-1993.)
Assertion
Ref Expression
sbco3 ([z / y][y / x]φ ↔ [z / x][x / y]φ)

Proof of Theorem sbco3
StepHypRef Expression
1 drsb1 2022 . . 3 (x x = y → ([z / x][y / x]φ ↔ [z / y][y / x]φ))
2 sbequ12a 1921 . . . . 5 (x = y → ([y / x]φ ↔ [x / y]φ))
32alimi 1559 . . . 4 (x x = yx([y / x]φ ↔ [x / y]φ))
4 spsbbi 2077 . . . 4 (x([y / x]φ ↔ [x / y]φ) → ([z / x][y / x]φ ↔ [z / x][x / y]φ))
53, 4syl 15 . . 3 (x x = y → ([z / x][y / x]φ ↔ [z / x][x / y]φ))
61, 5bitr3d 246 . 2 (x x = y → ([z / y][y / x]φ ↔ [z / x][x / y]φ))
7 sbco 2083 . . . 4 ([x / y][y / x]φ ↔ [x / y]φ)
87sbbii 1653 . . 3 ([z / x][x / y][y / x]φ ↔ [z / x][x / y]φ)
9 nfnae 1956 . . . 4 y ¬ x x = y
10 nfnae 1956 . . . 4 x ¬ x x = y
11 nfsb2 2058 . . . 4 x x = y → Ⅎx[y / x]φ)
129, 10, 11sbco2d 2087 . . 3 x x = y → ([z / x][x / y][y / x]φ ↔ [z / y][y / x]φ))
138, 12syl5rbbr 251 . 2 x x = y → ([z / y][y / x]φ ↔ [z / x][x / y]φ))
146, 13pm2.61i 156 1 ([z / y][y / x]φ ↔ [z / x][x / y]φ)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   ↔ wb 176  ∀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: (None)
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