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Theorem sbcom2 2114
 Description: Commutativity law for substitution. Used in proof of Theorem 9.7 of [Megill] p. 449 (p. 16 of the preprint). (Contributed by NM, 27-May-1997.)
Assertion
Ref Expression
sbcom2 ([w / z][y / x]φ ↔ [y / x][w / z]φ)
Distinct variable groups:   x,z   x,w   y,z
Allowed substitution hints:   φ(x,y,z,w)

Proof of Theorem sbcom2
StepHypRef Expression
1 alcom 1737 . . . . . 6 (zx(x = y → (z = wφ)) ↔ xz(x = y → (z = wφ)))
2 bi2.04 350 . . . . . . . . 9 ((x = y → (z = wφ)) ↔ (z = w → (x = yφ)))
32albii 1566 . . . . . . . 8 (x(x = y → (z = wφ)) ↔ x(z = w → (x = yφ)))
4 19.21v 1890 . . . . . . . 8 (x(z = w → (x = yφ)) ↔ (z = wx(x = yφ)))
53, 4bitri 240 . . . . . . 7 (x(x = y → (z = wφ)) ↔ (z = wx(x = yφ)))
65albii 1566 . . . . . 6 (zx(x = y → (z = wφ)) ↔ z(z = wx(x = yφ)))
7 19.21v 1890 . . . . . . 7 (z(x = y → (z = wφ)) ↔ (x = yz(z = wφ)))
87albii 1566 . . . . . 6 (xz(x = y → (z = wφ)) ↔ x(x = yz(z = wφ)))
91, 6, 83bitr3i 266 . . . . 5 (z(z = wx(x = yφ)) ↔ x(x = yz(z = wφ)))
109a1i 10 . . . 4 ((¬ x x = y ¬ z z = w) → (z(z = wx(x = yφ)) ↔ x(x = yz(z = wφ))))
11 sb4b 2054 . . . . 5 z z = w → ([w / z][y / x]φz(z = w → [y / x]φ)))
12 sb4b 2054 . . . . . . 7 x x = y → ([y / x]φx(x = yφ)))
1312imbi2d 307 . . . . . 6 x x = y → ((z = w → [y / x]φ) ↔ (z = wx(x = yφ))))
1413albidv 1625 . . . . 5 x x = y → (z(z = w → [y / x]φ) ↔ z(z = wx(x = yφ))))
1511, 14sylan9bbr 681 . . . 4 ((¬ x x = y ¬ z z = w) → ([w / z][y / x]φz(z = wx(x = yφ))))
16 sb4b 2054 . . . . 5 x x = y → ([y / x][w / z]φx(x = y → [w / z]φ)))
17 sb4b 2054 . . . . . . 7 z z = w → ([w / z]φz(z = wφ)))
1817imbi2d 307 . . . . . 6 z z = w → ((x = y → [w / z]φ) ↔ (x = yz(z = wφ))))
1918albidv 1625 . . . . 5 z z = w → (x(x = y → [w / z]φ) ↔ x(x = yz(z = wφ))))
2016, 19sylan9bb 680 . . . 4 ((¬ x x = y ¬ z z = w) → ([y / x][w / z]φx(x = yz(z = wφ))))
2110, 15, 203bitr4d 276 . . 3 ((¬ x x = y ¬ z z = w) → ([w / z][y / x]φ ↔ [y / x][w / z]φ))
2221ex 423 . 2 x x = y → (¬ z z = w → ([w / z][y / x]φ ↔ [y / x][w / z]φ)))
23 nfae 1954 . . . 4 zx x = y
24 sbequ12 1919 . . . . 5 (x = y → (φ ↔ [y / x]φ))
2524sps 1754 . . . 4 (x x = y → (φ ↔ [y / x]φ))
2623, 25sbbid 2078 . . 3 (x x = y → ([w / z]φ ↔ [w / z][y / x]φ))
27 sbequ12 1919 . . . 4 (x = y → ([w / z]φ ↔ [y / x][w / z]φ))
2827sps 1754 . . 3 (x x = y → ([w / z]φ ↔ [y / x][w / z]φ))
2926, 28bitr3d 246 . 2 (x x = y → ([w / z][y / x]φ ↔ [y / x][w / z]φ))
30 sbequ12 1919 . . . 4 (z = w → ([y / x]φ ↔ [w / z][y / x]φ))
3130sps 1754 . . 3 (z z = w → ([y / x]φ ↔ [w / z][y / x]φ))
32 nfae 1954 . . . 4 xz z = w
33 sbequ12 1919 . . . . 5 (z = w → (φ ↔ [w / z]φ))
3433sps 1754 . . . 4 (z z = w → (φ ↔ [w / z]φ))
3532, 34sbbid 2078 . . 3 (z z = w → ([y / x]φ ↔ [y / x][w / z]φ))
3631, 35bitr3d 246 . 2 (z z = w → ([w / z][y / x]φ ↔ [y / x][w / z]φ))
3722, 29, 36pm2.61ii 157 1 ([w / z][y / x]φ ↔ [y / x][w / z]φ)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 176   ∧ wa 358  ∀wal 1540  [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-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649 This theorem is referenced by:  2sb5rf  2117  2sb6rf  2118  2eu6  2289
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