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Theorem drsb1 2365
 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, 2-Jun-1993.)
Assertion
Ref Expression
drsb1 (∀𝑥 𝑥 = 𝑦 → ([𝑧 / 𝑥]𝜑 ↔ [𝑧 / 𝑦]𝜑))

Proof of Theorem drsb1
StepHypRef Expression
1 equequ1 1939 . . . . 5 (𝑥 = 𝑦 → (𝑥 = 𝑧𝑦 = 𝑧))
21sps 2043 . . . 4 (∀𝑥 𝑥 = 𝑦 → (𝑥 = 𝑧𝑦 = 𝑧))
32imbi1d 330 . . 3 (∀𝑥 𝑥 = 𝑦 → ((𝑥 = 𝑧𝜑) ↔ (𝑦 = 𝑧𝜑)))
42anbi1d 737 . . . 4 (∀𝑥 𝑥 = 𝑦 → ((𝑥 = 𝑧𝜑) ↔ (𝑦 = 𝑧𝜑)))
54drex1 2315 . . 3 (∀𝑥 𝑥 = 𝑦 → (∃𝑥(𝑥 = 𝑧𝜑) ↔ ∃𝑦(𝑦 = 𝑧𝜑)))
63, 5anbi12d 743 . 2 (∀𝑥 𝑥 = 𝑦 → (((𝑥 = 𝑧𝜑) ∧ ∃𝑥(𝑥 = 𝑧𝜑)) ↔ ((𝑦 = 𝑧𝜑) ∧ ∃𝑦(𝑦 = 𝑧𝜑))))
7 df-sb 1868 . 2 ([𝑧 / 𝑥]𝜑 ↔ ((𝑥 = 𝑧𝜑) ∧ ∃𝑥(𝑥 = 𝑧𝜑)))
8 df-sb 1868 . 2 ([𝑧 / 𝑦]𝜑 ↔ ((𝑦 = 𝑧𝜑) ∧ ∃𝑦(𝑦 = 𝑧𝜑)))
96, 7, 83bitr4g 302 1 (∀𝑥 𝑥 = 𝑦 → ([𝑧 / 𝑥]𝜑 ↔ [𝑧 / 𝑦]𝜑))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383  ∀wal 1473  ∃wex 1695  [wsb 1867 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-12 2034  ax-13 2234 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868 This theorem is referenced by:  sbco3  2405  iotaeq  5776
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