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Theorem drsb1 1677
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 1595 . . . . 5 (x = y → (x = zy = z))
21sps 1427 . . . 4 (x x = y → (x = zy = z))
32imbi1d 220 . . 3 (x x = y → ((x = zφ) ↔ (y = zφ)))
42anbi1d 438 . . . 4 (x x = y → ((x = z φ) ↔ (y = z φ)))
54drex1 1676 . . 3 (x x = y → (x(x = z φ) ↔ y(y = z φ)))
63, 5anbi12d 442 . 2 (x x = y → (((x = zφ) x(x = z φ)) ↔ ((y = zφ) y(y = z φ))))
7 df-sb 1643 . 2 ([z / x]φ ↔ ((x = zφ) x(x = z φ)))
8 df-sb 1643 . 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 1240  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-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-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:  sbequi  1717  nfsbxy  1815  nfsbxyt  1816  sbcomxyyz  1843  iotaeq  4818
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