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

Theorem bj-alequexv 32780
Description: Version of bj-alequex 32833 with DV(x,y), requiring fewer axioms. (Contributed by BJ, 9-Nov-2021.)
Assertion
Ref Expression
bj-alequexv (∀𝑥(𝑥 = 𝑦𝜑) → ∃𝑥𝜑)
Distinct variable group:   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem bj-alequexv
StepHypRef Expression
1 ax6ev 1947 . 2 𝑥 𝑥 = 𝑦
2 exim 1801 . 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 1521  wex 1744
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-6 1945
This theorem depends on definitions:  df-bi 197  df-ex 1745
This theorem is referenced by:  bj-spimvwt  32781
  Copyright terms: Public domain W3C validator