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Theorem sbel2x 2125
 Description: Elimination of double substitution. (Contributed by NM, 5-Aug-1993.)
Assertion
Ref Expression
sbel2x (φxy((x = z y = w) [y / w][x / z]φ))
Distinct variable groups:   x,y,z   y,w   φ,x,y
Allowed substitution hints:   φ(z,w)

Proof of Theorem sbel2x
StepHypRef Expression
1 sbelx 2124 . . . . 5 ([x / z]φy(y = w [y / w][x / z]φ))
21anbi2i 675 . . . 4 ((x = z [x / z]φ) ↔ (x = z y(y = w [y / w][x / z]φ)))
32exbii 1582 . . 3 (x(x = z [x / z]φ) ↔ x(x = z y(y = w [y / w][x / z]φ)))
4 sbelx 2124 . . 3 (φx(x = z [x / z]φ))
5 exdistr 1906 . . 3 (xy(x = z (y = w [y / w][x / z]φ)) ↔ x(x = z y(y = w [y / w][x / z]φ)))
63, 4, 53bitr4i 268 . 2 (φxy(x = z (y = w [y / w][x / z]φ)))
7 anass 630 . . 3 (((x = z y = w) [y / w][x / z]φ) ↔ (x = z (y = w [y / w][x / z]φ)))
872exbii 1583 . 2 (xy((x = z y = w) [y / w][x / z]φ) ↔ xy(x = z (y = w [y / w][x / z]φ)))
96, 8bitr4i 243 1 (φxy((x = z y = w) [y / w][x / 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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