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

Theorem bj-chvarv 32420
Description: Version of chvar 2261 with a dv condition, which does not require ax-13 2245. (Contributed by BJ, 31-May-2019.) (Proof modification is discouraged.)
Hypotheses
Ref Expression
bj-chvarv.nf 𝑥𝜓
bj-chvarv.1 (𝑥 = 𝑦 → (𝜑𝜓))
bj-chvarv.2 𝜑
Assertion
Ref Expression
bj-chvarv 𝜓
Distinct variable group:   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑥,𝑦)

Proof of Theorem bj-chvarv
StepHypRef Expression
1 bj-chvarv.nf . . 3 𝑥𝜓
2 bj-chvarv.1 . . . 4 (𝑥 = 𝑦 → (𝜑𝜓))
32biimpd 219 . . 3 (𝑥 = 𝑦 → (𝜑𝜓))
41, 3spimv1 2112 . 2 (∀𝑥𝜑𝜓)
5 bj-chvarv.2 . 2 𝜑
64, 5mpg 1721 1 𝜓
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wnf 1705
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-12 2044
This theorem depends on definitions:  df-bi 197  df-ex 1702  df-nf 1707
This theorem is referenced by:  bj-axrep2  32485  bj-axrep3  32486
  Copyright terms: Public domain W3C validator