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

Theorem bj-alequex 34182
Description: A fol lemma. See alequexv 2007 for a version with a disjoint variable condition requiring fewer axioms. Can be used to reduce the proof of spimt 2405 from 133 to 112 bytes. (Contributed by BJ, 6-Oct-2018.)
Assertion
Ref Expression
bj-alequex (∀𝑥(𝑥 = 𝑦𝜑) → ∃𝑥𝜑)

Proof of Theorem bj-alequex
StepHypRef Expression
1 ax6e 2402 . 2 𝑥 𝑥 = 𝑦
2 exim 1835 . 2 (∀𝑥(𝑥 = 𝑦𝜑) → (∃𝑥 𝑥 = 𝑦 → ∃𝑥𝜑))
31, 2mpi 20 1 (∀𝑥(𝑥 = 𝑦𝜑) → ∃𝑥𝜑)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1536  wex 1781
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-12 2178  ax-13 2391
This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782
This theorem is referenced by:  bj-spimt2  34183  bj-equsal1t  34221
  Copyright terms: Public domain W3C validator