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Theorem pm11.07 2115
 Description: Theorem *11.07 in [WhiteheadRussell] p. 159. (Contributed by Andrew Salmon, 17-Jun-2011.)
Assertion
Ref Expression
pm11.07 ([w / x][y / z]φ ↔ [y / x][w / z]φ)
Distinct variable groups:   φ,x,y,z   x,w,z
Allowed substitution hint:   φ(w)

Proof of Theorem pm11.07
StepHypRef Expression
1 a9ev 1656 . . . . . . 7 x x = w
2 a9ev 1656 . . . . . . 7 z z = y
31, 2pm3.2i 441 . . . . . 6 (x x = w z z = y)
4 a9ev 1656 . . . . . . 7 x x = y
5 a9ev 1656 . . . . . . 7 z z = w
64, 5pm3.2i 441 . . . . . 6 (x x = y z z = w)
73, 62th 230 . . . . 5 ((x x = w z z = y) ↔ (x x = y z z = w))
8 eeanv 1913 . . . . 5 (xz(x = w z = y) ↔ (x x = w z z = y))
9 eeanv 1913 . . . . 5 (xz(x = y z = w) ↔ (x x = y z z = w))
107, 8, 93bitr4i 268 . . . 4 (xz(x = w z = y) ↔ xz(x = y z = w))
1110anbi1i 676 . . 3 ((xz(x = w z = y) φ) ↔ (xz(x = y z = w) φ))
12 19.41vv 1902 . . 3 (xz((x = w z = y) φ) ↔ (xz(x = w z = y) φ))
13 19.41vv 1902 . . 3 (xz((x = y z = w) φ) ↔ (xz(x = y z = w) φ))
1411, 12, 133bitr4i 268 . 2 (xz((x = w z = y) φ) ↔ xz((x = y z = w) φ))
15 2sb5 2112 . 2 ([w / x][y / z]φxz((x = w z = y) φ))
16 2sb5 2112 . 2 ([y / x][w / z]φxz((x = y z = w) φ))
1714, 15, 163bitr4i 268 1 ([w / x][y / z]φ ↔ [y / x][w / z]φ)
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 176   ∧ wa 358  ∃wex 1541  [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: (None)
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