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Theorem drsb1 2375
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 1950 . . . . 5 (𝑥 = 𝑦 → (𝑥 = 𝑧𝑦 = 𝑧))
21sps 2053 . . . 4 (∀𝑥 𝑥 = 𝑦 → (𝑥 = 𝑧𝑦 = 𝑧))
32imbi1d 331 . . 3 (∀𝑥 𝑥 = 𝑦 → ((𝑥 = 𝑧𝜑) ↔ (𝑦 = 𝑧𝜑)))
42anbi1d 740 . . . 4 (∀𝑥 𝑥 = 𝑦 → ((𝑥 = 𝑧𝜑) ↔ (𝑦 = 𝑧𝜑)))
54drex1 2325 . . 3 (∀𝑥 𝑥 = 𝑦 → (∃𝑥(𝑥 = 𝑧𝜑) ↔ ∃𝑦(𝑦 = 𝑧𝜑)))
63, 5anbi12d 746 . 2 (∀𝑥 𝑥 = 𝑦 → (((𝑥 = 𝑧𝜑) ∧ ∃𝑥(𝑥 = 𝑧𝜑)) ↔ ((𝑦 = 𝑧𝜑) ∧ ∃𝑦(𝑦 = 𝑧𝜑))))
7 df-sb 1879 . 2 ([𝑧 / 𝑥]𝜑 ↔ ((𝑥 = 𝑧𝜑) ∧ ∃𝑥(𝑥 = 𝑧𝜑)))
8 df-sb 1879 . 2 ([𝑧 / 𝑦]𝜑 ↔ ((𝑦 = 𝑧𝜑) ∧ ∃𝑦(𝑦 = 𝑧𝜑)))
96, 7, 83bitr4g 303 1 (∀𝑥 𝑥 = 𝑦 → ([𝑧 / 𝑥]𝜑 ↔ [𝑧 / 𝑦]𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  wal 1479  wex 1702  [wsb 1878
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-10 2017  ax-12 2045  ax-13 2244
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-ex 1703  df-nf 1708  df-sb 1879
This theorem is referenced by:  sbco3  2415  iotaeq  5847
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