Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  sbel2x Structured version   Visualization version   GIF version

Theorem sbel2x 2487
 Description: Elimination of double substitution. Usage of this theorem is discouraged because it depends on ax-13 2379. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 29-Sep-2018.) (New usage is discouraged.)
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 1915 . . 3 𝑦𝜑
2 nfv 1915 . . 3 𝑥𝜑
31, 22sb5rf 2485 . 2 (𝜑 ↔ ∃𝑦𝑥((𝑦 = 𝑤𝑥 = 𝑧) ∧ [𝑦 / 𝑤][𝑥 / 𝑧]𝜑))
4 ancom 464 . . . 4 ((𝑦 = 𝑤𝑥 = 𝑧) ↔ (𝑥 = 𝑧𝑦 = 𝑤))
54anbi1i 626 . . 3 (((𝑦 = 𝑤𝑥 = 𝑧) ∧ [𝑦 / 𝑤][𝑥 / 𝑧]𝜑) ↔ ((𝑥 = 𝑧𝑦 = 𝑤) ∧ [𝑦 / 𝑤][𝑥 / 𝑧]𝜑))
652exbii 1850 . 2 (∃𝑦𝑥((𝑦 = 𝑤𝑥 = 𝑧) ∧ [𝑦 / 𝑤][𝑥 / 𝑧]𝜑) ↔ ∃𝑦𝑥((𝑥 = 𝑧𝑦 = 𝑤) ∧ [𝑦 / 𝑤][𝑥 / 𝑧]𝜑))
7 excom 2166 . 2 (∃𝑦𝑥((𝑥 = 𝑧𝑦 = 𝑤) ∧ [𝑦 / 𝑤][𝑥 / 𝑧]𝜑) ↔ ∃𝑥𝑦((𝑥 = 𝑧𝑦 = 𝑤) ∧ [𝑦 / 𝑤][𝑥 / 𝑧]𝜑))
83, 6, 73bitri 300 1 (𝜑 ↔ ∃𝑥𝑦((𝑥 = 𝑧𝑦 = 𝑤) ∧ [𝑦 / 𝑤][𝑥 / 𝑧]𝜑))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 209   ∧ wa 399  ∃wex 1781  [wsb 2069 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-10 2142  ax-11 2158  ax-12 2175  ax-13 2379 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator