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

Theorem bj-spvv 32418
 Description: Version of spv 2259 with a dv condition, which does not require ax-7 1932, ax-12 2044, ax-13 2245. (Contributed by BJ, 31-May-2019.) (Proof modification is discouraged.)
Hypothesis
Ref Expression
bj-spvv.1 (𝑥 = 𝑦 → (𝜑𝜓))
Assertion
Ref Expression
bj-spvv (∀𝑥𝜑𝜓)
Distinct variable groups:   𝑥,𝑦   𝜓,𝑥
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑦)

Proof of Theorem bj-spvv
StepHypRef Expression
1 bj-spvv.1 . . 3 (𝑥 = 𝑦 → (𝜑𝜓))
21biimpd 219 . 2 (𝑥 = 𝑦 → (𝜑𝜓))
32bj-spimvv 32416 1 (∀𝑥𝜑𝜓)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196  ∀wal 1478 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 This theorem depends on definitions:  df-bi 197  df-ex 1702 This theorem is referenced by:  bj-chvarvv  32421  bj-nalset  32490  bj-ru0  32632
 Copyright terms: Public domain W3C validator