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Theorem sbal1 1875
 Description: A theorem used in elimination of disjoint variable restriction on x and y by replacing it with a distinctor ¬ ∀xx = z. (Contributed by NM, 5-Aug-1993.) (Proof rewitten by Jim Kingdon, 24-Feb-2018.)
Assertion
Ref Expression
sbal1 x x = z → ([z / y]xφx[z / y]φ))
Distinct variable group:   x,y
Allowed substitution hints:   φ(x,y,z)

Proof of Theorem sbal1
Dummy variable w is distinct from all other variables.
StepHypRef Expression
1 sbal 1873 . . . 4 ([w / y]xφx[w / y]φ)
21sbbii 1645 . . 3 ([z / w][w / y]xφ ↔ [z / w]x[w / y]φ)
3 sbal1yz 1874 . . 3 x x = z → ([z / w]x[w / y]φx[z / w][w / y]φ))
42, 3syl5bb 181 . 2 x x = z → ([z / w][w / y]xφx[z / w][w / y]φ))
5 ax-17 1416 . . 3 (xφwxφ)
65sbco2v 1818 . 2 ([z / w][w / y]xφ ↔ [z / y]xφ)
7 ax-17 1416 . . . 4 (φwφ)
87sbco2v 1818 . . 3 ([z / w][w / y]φ ↔ [z / y]φ)
98albii 1356 . 2 (x[z / w][w / y]φx[z / y]φ)
104, 6, 93bitr3g 211 1 x x = z → ([z / y]xφx[z / y]φ))
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 98  ∀wal 1240  [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-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425 This theorem depends on definitions:  df-bi 110  df-nf 1347  df-sb 1643 This theorem is referenced by: (None)
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