MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  spimv1 Structured version   Visualization version   GIF version

Theorem spimv1 2278
Description: Version of spim 2416 with a dv condition, which does not require ax-13 2408. See spimvw 2085 for a version with two dv conditions, requiring fewer axioms, and spimv 2419 for another variant. (Contributed by BJ, 31-May-2019.)
Hypotheses
Ref Expression
spimv1.nf 𝑥𝜓
spimv1.1 (𝑥 = 𝑦 → (𝜑𝜓))
Assertion
Ref Expression
spimv1 (∀𝑥𝜑𝜓)
Distinct variable group:   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑥,𝑦)

Proof of Theorem spimv1
StepHypRef Expression
1 spimv1.nf . 2 𝑥𝜓
2 ax6ev 2059 . . 3 𝑥 𝑥 = 𝑦
3 spimv1.1 . . 3 (𝑥 = 𝑦 → (𝜑𝜓))
42, 3eximii 1912 . 2 𝑥(𝜑𝜓)
51, 419.36i 2255 1 (∀𝑥𝜑𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1629  wnf 1856
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-12 2203
This theorem depends on definitions:  df-bi 197  df-ex 1853  df-nf 1858
This theorem is referenced by:  cbv3v  2333  cbv3v2  2334  bj-chvarv  33062
  Copyright terms: Public domain W3C validator