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Theorem drsb1 1662
Description: Formula-building lemma for use with the Distinctor Reduction Theorem. Part of Theorem 9.4 of [Megill] p. 448 (p. 16 of preprint). (Contributed by NM, 5-Aug-1993.)
Assertion
Ref Expression
drsb1 (x x = y → ([z / x]φ ↔ [z / y]φ))

Proof of Theorem drsb1
StepHypRef Expression
1 equequ1 1580 . . . . 5 (x = y → (x = zy = z))
21sps 1412 . . . 4 (x x = y → (x = zy = z))
32imbi1d 220 . . 3 (x x = y → ((x = zφ) ↔ (y = zφ)))
42anbi1d 441 . . . 4 (x x = y → ((x = z φ) ↔ (y = z φ)))
54drex1 1661 . . 3 (x x = y → (x(x = z φ) ↔ y(y = z φ)))
63, 5anbi12d 445 . 2 (x x = y → (((x = zφ) x(x = z φ)) ↔ ((y = zφ) y(y = z φ))))
7 df-sb 1628 . 2 ([z / x]φ ↔ ((x = zφ) x(x = z φ)))
8 df-sb 1628 . 2 ([z / y]φ ↔ ((y = zφ) y(y = z φ)))
96, 7, 83bitr4g 212 1 (x x = y → ([z / x]φ ↔ [z / y]φ))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98  wal 1226  wex 1362  [wsb 1627
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-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-4 1381  ax-17 1400  ax-i9 1404  ax-ial 1409
This theorem depends on definitions:  df-bi 110  df-sb 1628
This theorem is referenced by:  sbequi  1702  nfsbxy  1800  nfsbxyt  1801  sbcomxyyz  1828  iotaeq  4802
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