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Theorem drsb1 1792
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 1705 . . . . 5 (𝑥 = 𝑦 → (𝑥 = 𝑧𝑦 = 𝑧))
21sps 1530 . . . 4 (∀𝑥 𝑥 = 𝑦 → (𝑥 = 𝑧𝑦 = 𝑧))
32imbi1d 230 . . 3 (∀𝑥 𝑥 = 𝑦 → ((𝑥 = 𝑧𝜑) ↔ (𝑦 = 𝑧𝜑)))
42anbi1d 462 . . . 4 (∀𝑥 𝑥 = 𝑦 → ((𝑥 = 𝑧𝜑) ↔ (𝑦 = 𝑧𝜑)))
54drex1 1791 . . 3 (∀𝑥 𝑥 = 𝑦 → (∃𝑥(𝑥 = 𝑧𝜑) ↔ ∃𝑦(𝑦 = 𝑧𝜑)))
63, 5anbi12d 470 . 2 (∀𝑥 𝑥 = 𝑦 → (((𝑥 = 𝑧𝜑) ∧ ∃𝑥(𝑥 = 𝑧𝜑)) ↔ ((𝑦 = 𝑧𝜑) ∧ ∃𝑦(𝑦 = 𝑧𝜑))))
7 df-sb 1756 . 2 ([𝑧 / 𝑥]𝜑 ↔ ((𝑥 = 𝑧𝜑) ∧ ∃𝑥(𝑥 = 𝑧𝜑)))
8 df-sb 1756 . 2 ([𝑧 / 𝑦]𝜑 ↔ ((𝑦 = 𝑧𝜑) ∧ ∃𝑦(𝑦 = 𝑧𝜑)))
96, 7, 83bitr4g 222 1 (∀𝑥 𝑥 = 𝑦 → ([𝑧 / 𝑥]𝜑 ↔ [𝑧 / 𝑦]𝜑))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104  wal 1346  wex 1485  [wsb 1755
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527
This theorem depends on definitions:  df-bi 116  df-sb 1756
This theorem is referenced by:  sbequi  1832  nfsbxy  1935  nfsbxyt  1936  sbcomxyyz  1965  iotaeq  5168
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