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Theorem sbco2 2086
 Description: A composition law for substitution. (Contributed by NM, 30-Jun-1994.) (Revised by Mario Carneiro, 6-Oct-2016.)
Hypothesis
Ref Expression
sbco2.1 zφ
Assertion
Ref Expression
sbco2 ([y / z][z / x]φ ↔ [y / x]φ)

Proof of Theorem sbco2
StepHypRef Expression
1 sbco2.1 . . . . . 6 zφ
21sbid2 2084 . . . . 5 ([x / z][z / x]φφ)
3 sbequ 2060 . . . . 5 (x = y → ([x / z][z / x]φ ↔ [y / z][z / x]φ))
42, 3syl5bbr 250 . . . 4 (x = y → (φ ↔ [y / z][z / x]φ))
5 sbequ12 1919 . . . 4 (x = y → (φ ↔ [y / x]φ))
64, 5bitr3d 246 . . 3 (x = y → ([y / z][z / x]φ ↔ [y / x]φ))
76sps 1754 . 2 (x x = y → ([y / z][z / x]φ ↔ [y / x]φ))
8 nfnae 1956 . . . 4 x ¬ x x = y
91nfs1 2044 . . . . 5 x[z / x]φ
109nfsb4 2081 . . . 4 x x = y → Ⅎx[y / z][z / x]φ)
114a1i 10 . . . 4 x x = y → (x = y → (φ ↔ [y / z][z / x]φ)))
128, 10, 11sbied 2036 . . 3 x x = y → ([y / x]φ ↔ [y / z][z / x]φ))
1312bicomd 192 . 2 x x = y → ([y / z][z / x]φ ↔ [y / x]φ))
147, 13pm2.61i 156 1 ([y / z][z / x]φ ↔ [y / x]φ)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 176  ∀wal 1540  Ⅎwnf 1544  [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-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649 This theorem is referenced by:  sbco2d  2087  equsb3  2102  elsb3  2103  elsb4  2104  dfsb7  2119  sb7f  2120  2eu6  2289  eqsb3  2454  clelsb3  2455  sbralie  2848  sbcco  3068
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