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Theorem drsb1 1755
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 1673 . . . . 5 (𝑥 = 𝑦 → (𝑥 = 𝑧𝑦 = 𝑧))
21sps 1502 . . . 4 (∀𝑥 𝑥 = 𝑦 → (𝑥 = 𝑧𝑦 = 𝑧))
32imbi1d 230 . . 3 (∀𝑥 𝑥 = 𝑦 → ((𝑥 = 𝑧𝜑) ↔ (𝑦 = 𝑧𝜑)))
42anbi1d 460 . . . 4 (∀𝑥 𝑥 = 𝑦 → ((𝑥 = 𝑧𝜑) ↔ (𝑦 = 𝑧𝜑)))
54drex1 1754 . . 3 (∀𝑥 𝑥 = 𝑦 → (∃𝑥(𝑥 = 𝑧𝜑) ↔ ∃𝑦(𝑦 = 𝑧𝜑)))
63, 5anbi12d 464 . 2 (∀𝑥 𝑥 = 𝑦 → (((𝑥 = 𝑧𝜑) ∧ ∃𝑥(𝑥 = 𝑧𝜑)) ↔ ((𝑦 = 𝑧𝜑) ∧ ∃𝑦(𝑦 = 𝑧𝜑))))
7 df-sb 1721 . 2 ([𝑧 / 𝑥]𝜑 ↔ ((𝑥 = 𝑧𝜑) ∧ ∃𝑥(𝑥 = 𝑧𝜑)))
8 df-sb 1721 . 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 1314  wex 1453  [wsb 1720
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 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499
This theorem depends on definitions:  df-bi 116  df-sb 1721
This theorem is referenced by:  sbequi  1795  nfsbxy  1895  nfsbxyt  1896  sbcomxyyz  1923  iotaeq  5066
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