Users' Mathboxes Mathbox for Wolf Lammen < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  wl-2spsbbi Structured version   Visualization version   GIF version

Theorem wl-2spsbbi 34848
Description: spsbbi 2079 applied twice. (Contributed by Wolf Lammen, 5-Aug-2023.)
Assertion
Ref Expression
wl-2spsbbi (∀𝑎𝑏(𝜑𝜓) → ([𝑦 / 𝑏][𝑥 / 𝑎]𝜑 ↔ [𝑦 / 𝑏][𝑥 / 𝑎]𝜓))

Proof of Theorem wl-2spsbbi
StepHypRef Expression
1 alcom 2164 . 2 (∀𝑎𝑏(𝜑𝜓) ↔ ∀𝑏𝑎(𝜑𝜓))
2 nfa1 2156 . . 3 𝑏𝑏𝑎(𝜑𝜓)
3 nfa1 2156 . . . . 5 𝑎𝑎(𝜑𝜓)
4 sp 2183 . . . . 5 (∀𝑎(𝜑𝜓) → (𝜑𝜓))
53, 4sbbid 2247 . . . 4 (∀𝑎(𝜑𝜓) → ([𝑥 / 𝑎]𝜑 ↔ [𝑥 / 𝑎]𝜓))
65sps 2185 . . 3 (∀𝑏𝑎(𝜑𝜓) → ([𝑥 / 𝑎]𝜑 ↔ [𝑥 / 𝑎]𝜓))
72, 6sbbid 2247 . 2 (∀𝑏𝑎(𝜑𝜓) → ([𝑦 / 𝑏][𝑥 / 𝑎]𝜑 ↔ [𝑦 / 𝑏][𝑥 / 𝑎]𝜓))
81, 7sylbi 220 1 (∀𝑎𝑏(𝜑𝜓) → ([𝑦 / 𝑏][𝑥 / 𝑎]𝜑 ↔ [𝑦 / 𝑏][𝑥 / 𝑎]𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wal 1536  [wsb 2070
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 1912  ax-6 1971  ax-7 2016  ax-10 2146  ax-11 2162  ax-12 2178
This theorem depends on definitions:  df-bi 210  df-or 845  df-ex 1782  df-nf 1786  df-sb 2071
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator