Intuitionistic Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  ILE Home  >  Th. List  >  r19.29af GIF version

Theorem r19.29af 2470
 Description: A commonly used pattern based on r19.29 2467 (Contributed by Thierry Arnoux, 29-Nov-2017.)
Hypotheses
Ref Expression
r19.29af.0 𝑥𝜑
r19.29af.1 (((𝜑𝑥𝐴) ∧ 𝜓) → 𝜒)
r19.29af.2 (𝜑 → ∃𝑥𝐴 𝜓)
Assertion
Ref Expression
r19.29af (𝜑𝜒)
Distinct variable group:   𝜒,𝑥
Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑥)   𝐴(𝑥)

Proof of Theorem r19.29af
StepHypRef Expression
1 r19.29af.0 . 2 𝑥𝜑
2 nfv 1437 . 2 𝑥𝜒
3 r19.29af.1 . 2 (((𝜑𝑥𝐴) ∧ 𝜓) → 𝜒)
4 r19.29af.2 . 2 (𝜑 → ∃𝑥𝐴 𝜓)
51, 2, 3, 4r19.29af2 2469 1 (𝜑𝜒)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 101  Ⅎwnf 1365   ∈ wcel 1409  ∃wrex 2324 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-5 1352  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-4 1416  ax-17 1435  ax-ial 1443  ax-i5r 1444 This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-ral 2328  df-rex 2329 This theorem is referenced by:  r19.29a  2471
 Copyright terms: Public domain W3C validator