Users' Mathboxes Mathbox for BJ < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  bj-hbs1 Structured version   Visualization version   GIF version

Theorem bj-hbs1 35035
Description: Version of hbsb2 2484 with a disjoint variable condition, which does not require ax-13 2370, and removal of ax-13 2370 from hbs1 2264. (Contributed by BJ, 23-Jun-2019.) (Proof modification is discouraged.)
Assertion
Ref Expression
bj-hbs1 ([𝑦 / 𝑥]𝜑 → ∀𝑥[𝑦 / 𝑥]𝜑)
Distinct variable group:   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem bj-hbs1
StepHypRef Expression
1 sb6 2086 . 2 ([𝑦 / 𝑥]𝜑 ↔ ∀𝑥(𝑥 = 𝑦𝜑))
21biimpri 227 . . 3 (∀𝑥(𝑥 = 𝑦𝜑) → [𝑦 / 𝑥]𝜑)
32axc4i 2314 . 2 (∀𝑥(𝑥 = 𝑦𝜑) → ∀𝑥[𝑦 / 𝑥]𝜑)
41, 3sylbi 216 1 ([𝑦 / 𝑥]𝜑 → ∀𝑥[𝑦 / 𝑥]𝜑)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1537  [wsb 2065
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-10 2135  ax-12 2169
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 846  df-ex 1780  df-nf 1784  df-sb 2066
This theorem is referenced by:  bj-nfs1v  35036
  Copyright terms: Public domain W3C validator