MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  sb2ae Structured version   Visualization version   GIF version

Theorem sb2ae 2490
Description: In the case of two successive substitutions for two always equal variables, the second substitution has no effect. (Contributed by BJ and WL, 9-Aug-2023.)
Assertion
Ref Expression
sb2ae (∀𝑥 𝑥 = 𝑦 → ([𝑢 / 𝑥][𝑣 / 𝑦]𝜑 ↔ [𝑣 / 𝑦]𝜑))
Distinct variable group:   𝑦,𝑣
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑣,𝑢)

Proof of Theorem sb2ae
StepHypRef Expression
1 drsb1 2489 . 2 (∀𝑥 𝑥 = 𝑦 → ([𝑢 / 𝑥][𝑣 / 𝑦]𝜑 ↔ [𝑢 / 𝑦][𝑣 / 𝑦]𝜑))
2 nfs1v 2237 . . 3 𝑦[𝑣 / 𝑦]𝜑
32sbf 2234 . 2 ([𝑢 / 𝑦][𝑣 / 𝑦]𝜑 ↔ [𝑣 / 𝑦]𝜑)
41, 3syl6bb 288 1 (∀𝑥 𝑥 = 𝑦 → ([𝑢 / 𝑥][𝑣 / 𝑦]𝜑 ↔ [𝑣 / 𝑦]𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wal 1520  [wsb 2042
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-10 2112  ax-12 2141  ax-13 2344
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-ex 1762  df-nf 1766  df-sb 2043
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator