ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  r19.29af2 GIF version

Theorem r19.29af2 2501
Description: A commonly used pattern based on r19.29 2499 (Contributed by Thierry Arnoux, 17-Dec-2017.)
Hypotheses
Ref Expression
r19.29af2.p 𝑥𝜑
r19.29af2.c 𝑥𝜒
r19.29af2.1 (((𝜑𝑥𝐴) ∧ 𝜓) → 𝜒)
r19.29af2.2 (𝜑 → ∃𝑥𝐴 𝜓)
Assertion
Ref Expression
r19.29af2 (𝜑𝜒)

Proof of Theorem r19.29af2
StepHypRef Expression
1 r19.29af2.2 . . 3 (𝜑 → ∃𝑥𝐴 𝜓)
2 r19.29af2.p . . . 4 𝑥𝜑
3 r19.29af2.1 . . . . 5 (((𝜑𝑥𝐴) ∧ 𝜓) → 𝜒)
43exp31 356 . . . 4 (𝜑 → (𝑥𝐴 → (𝜓𝜒)))
52, 4ralrimi 2437 . . 3 (𝜑 → ∀𝑥𝐴 (𝜓𝜒))
61, 5jca 300 . 2 (𝜑 → (∃𝑥𝐴 𝜓 ∧ ∀𝑥𝐴 (𝜓𝜒)))
7 r19.29r 2500 . 2 ((∃𝑥𝐴 𝜓 ∧ ∀𝑥𝐴 (𝜓𝜒)) → ∃𝑥𝐴 (𝜓 ∧ (𝜓𝜒)))
8 r19.29af2.c . . 3 𝑥𝜒
9 pm3.35 339 . . . 4 ((𝜓 ∧ (𝜓𝜒)) → 𝜒)
109a1i 9 . . 3 (𝑥𝐴 → ((𝜓 ∧ (𝜓𝜒)) → 𝜒))
118, 10rexlimi 2475 . 2 (∃𝑥𝐴 (𝜓 ∧ (𝜓𝜒)) → 𝜒)
126, 7, 113syl 17 1 (𝜑𝜒)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wnf 1390  wcel 1434  wral 2353  wrex 2354
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1377  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-4 1441  ax-17 1460  ax-ial 1468  ax-i5r 1469
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-ral 2358  df-rex 2359
This theorem is referenced by:  r19.29af  2502
  Copyright terms: Public domain W3C validator