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Theorem sbel2x 1871
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 1870 . . . . 5 ([x / z]φy(y = w [y / w][x / z]φ))
21anbi2i 430 . . . 4 ((x = z [x / z]φ) ↔ (x = z y(y = w [y / w][x / z]φ)))
32exbii 1493 . . 3 (x(x = z [x / z]φ) ↔ x(x = z y(y = w [y / w][x / z]φ)))
4 sbelx 1870 . . 3 (φx(x = z [x / z]φ))
5 exdistr 1784 . . 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 201 . 2 (φxy(x = z (y = w [y / w][x / z]φ)))
7 anass 381 . . 3 (((x = z y = w) [y / w][x / z]φ) ↔ (x = z (y = w [y / w][x / z]φ)))
872exbii 1494 . 2 (xy((x = z y = w) [y / w][x / z]φ) ↔ xy(x = z (y = w [y / w][x / z]φ)))
96, 8bitr4i 176 1 (φxy((x = z y = w) [y / w][x / z]φ))
Colors of variables: wff set class
Syntax hints:   wa 97  wb 98  wex 1378  [wsb 1642
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1333  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-11 1394  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424
This theorem depends on definitions:  df-bi 110  df-sb 1643
This theorem is referenced by: (None)
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