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Theorem drsb1 2022
 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 1684 . . . . 5 (x = y → (x = zy = z))
21sps 1754 . . . 4 (x x = y → (x = zy = z))
32imbi1d 308 . . 3 (x x = y → ((x = zφ) ↔ (y = zφ)))
42anbi1d 685 . . . 4 (x x = y → ((x = z φ) ↔ (y = z φ)))
54drex1 1967 . . 3 (x x = y → (x(x = z φ) ↔ y(y = z φ)))
63, 5anbi12d 691 . 2 (x x = y → (((x = zφ) x(x = z φ)) ↔ ((y = zφ) y(y = z φ))))
7 df-sb 1649 . 2 ([z / x]φ ↔ ((x = zφ) x(x = z φ)))
8 df-sb 1649 . 2 ([z / y]φ ↔ ((y = zφ) y(y = z φ)))
96, 7, 83bitr4g 279 1 (x x = y → ([z / x]φ ↔ [z / y]φ))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358  ∀wal 1540  ∃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:  sbequi  2059  nfsb4t  2080  sbco3  2088  sbcom  2089  sb9i  2094  iotaeq  4347
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