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Theorem sbel2x 2458
 Description: Elimination of double substitution. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 29-Sep-2018.)
Assertion
Ref Expression
sbel2x (𝜑 ↔ ∃𝑥𝑦((𝑥 = 𝑧𝑦 = 𝑤) ∧ [𝑦 / 𝑤][𝑥 / 𝑧]𝜑))
Distinct variable group:   𝑥,𝑦,𝜑
Allowed substitution hints:   𝜑(𝑧,𝑤)

Proof of Theorem sbel2x
StepHypRef Expression
1 nfv 1840 . . 3 𝑦𝜑
2 nfv 1840 . . 3 𝑥𝜑
31, 22sb5rf 2450 . 2 (𝜑 ↔ ∃𝑦𝑥((𝑦 = 𝑤𝑥 = 𝑧) ∧ [𝑦 / 𝑤][𝑥 / 𝑧]𝜑))
4 ancom 466 . . . 4 ((𝑦 = 𝑤𝑥 = 𝑧) ↔ (𝑥 = 𝑧𝑦 = 𝑤))
54anbi1i 730 . . 3 (((𝑦 = 𝑤𝑥 = 𝑧) ∧ [𝑦 / 𝑤][𝑥 / 𝑧]𝜑) ↔ ((𝑥 = 𝑧𝑦 = 𝑤) ∧ [𝑦 / 𝑤][𝑥 / 𝑧]𝜑))
652exbii 1772 . 2 (∃𝑦𝑥((𝑦 = 𝑤𝑥 = 𝑧) ∧ [𝑦 / 𝑤][𝑥 / 𝑧]𝜑) ↔ ∃𝑦𝑥((𝑥 = 𝑧𝑦 = 𝑤) ∧ [𝑦 / 𝑤][𝑥 / 𝑧]𝜑))
7 excom 2039 . 2 (∃𝑦𝑥((𝑥 = 𝑧𝑦 = 𝑤) ∧ [𝑦 / 𝑤][𝑥 / 𝑧]𝜑) ↔ ∃𝑥𝑦((𝑥 = 𝑧𝑦 = 𝑤) ∧ [𝑦 / 𝑤][𝑥 / 𝑧]𝜑))
83, 6, 73bitri 286 1 (𝜑 ↔ ∃𝑥𝑦((𝑥 = 𝑧𝑦 = 𝑤) ∧ [𝑦 / 𝑤][𝑥 / 𝑧]𝜑))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 196   ∧ wa 384  ∃wex 1701  [wsb 1877 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878 This theorem is referenced by: (None)
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