ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  chvarfv GIF version

Theorem chvarfv 1680
Description: Implicit substitution of 𝑦 for 𝑥 into a theorem. Version of chvar 1737 with a disjoint variable condition. (Contributed by Raph Levien, 9-Jul-2003.) (Revised by BJ, 31-May-2019.)
Hypotheses
Ref Expression
chvarfv.nf 𝑥𝜓
chvarfv.1 (𝑥 = 𝑦 → (𝜑𝜓))
chvarfv.2 𝜑
Assertion
Ref Expression
chvarfv 𝜓
Distinct variable group:   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑥,𝑦)

Proof of Theorem chvarfv
StepHypRef Expression
1 chvarfv.nf . . 3 𝑥𝜓
2 chvarfv.1 . . . 4 (𝑥 = 𝑦 → (𝜑𝜓))
32biimpd 143 . . 3 (𝑥 = 𝑦 → (𝜑𝜓))
41, 3spimfv 1679 . 2 (∀𝑥𝜑𝜓)
5 chvarfv.2 . 2 𝜑
64, 5mpg 1431 1 𝜓
Colors of variables: wff set class
Syntax hints:  wi 4  wb 104  wnf 1440
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1427  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-4 1490  ax-i9 1510  ax-ial 1514
This theorem depends on definitions:  df-bi 116  df-nf 1441
This theorem is referenced by:  fproddivapf  11528  fprodsplitf  11529
  Copyright terms: Public domain W3C validator