Users' Mathboxes Mathbox for Steven Nguyen < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  sbalexi Structured version   Visualization version   GIF version

Theorem sbalexi 42155
Description: Inference form of sbalex 2238, avoiding ax-10 2136 by using ax-gen 1793. (Contributed by SN, 12-Aug-2025.)
Hypothesis
Ref Expression
sbalexi.1 𝑥(𝑥 = 𝑦𝜑)
Assertion
Ref Expression
sbalexi 𝑥(𝑥 = 𝑦𝜑)
Distinct variable group:   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem sbalexi
StepHypRef Expression
1 sbalexi.1 . . 3 𝑥(𝑥 = 𝑦𝜑)
2 ax12ev2 2176 . . 3 (∃𝑥(𝑥 = 𝑦𝜑) → (𝑥 = 𝑦𝜑))
31, 2ax-mp 5 . 2 (𝑥 = 𝑦𝜑)
43ax-gen 1793 1 𝑥(𝑥 = 𝑦𝜑)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wal 1535  wex 1777
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-12 2173
This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1778
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator