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

Theorem bj-speiv 32849
Description: Version of spei 2297 with a dv condition, which does not require ax-13 2282 (neither ax-7 1981 nor ax-12 2087). (Contributed by BJ, 31-May-2019.) (Proof modification is discouraged.)
Hypotheses
Ref Expression
bj-speiv.1 (𝑥 = 𝑦 → (𝜑𝜓))
bj-speiv.2 𝜓
Assertion
Ref Expression
bj-speiv 𝑥𝜑
Distinct variable group:   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑥,𝑦)

Proof of Theorem bj-speiv
StepHypRef Expression
1 ax6ev 1947 . 2 𝑥 𝑥 = 𝑦
2 bj-speiv.2 . . 3 𝜓
3 bj-speiv.1 . . 3 (𝑥 = 𝑦 → (𝜑𝜓))
42, 3mpbiri 248 . 2 (𝑥 = 𝑦𝜑)
51, 4eximii 1804 1 𝑥𝜑
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  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: (None)
  Copyright terms: Public domain W3C validator