ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  r19.44mv GIF version

Theorem r19.44mv 3532
Description: Restricted version of Theorem 19.44 of [Margaris] p. 90. (Contributed by NM, 27-May-1998.)
Assertion
Ref Expression
r19.44mv (∃𝑦 𝑦𝐴 → (∃𝑥𝐴 (𝜑𝜓) ↔ (∃𝑥𝐴 𝜑𝜓)))
Distinct variable groups:   𝑥,𝐴   𝑦,𝐴   𝜓,𝑥
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑦)

Proof of Theorem r19.44mv
StepHypRef Expression
1 r19.43 2648 . 2 (∃𝑥𝐴 (𝜑𝜓) ↔ (∃𝑥𝐴 𝜑 ∨ ∃𝑥𝐴 𝜓))
2 r19.9rmv 3529 . . 3 (∃𝑦 𝑦𝐴 → (𝜓 ↔ ∃𝑥𝐴 𝜓))
32orbi2d 791 . 2 (∃𝑦 𝑦𝐴 → ((∃𝑥𝐴 𝜑𝜓) ↔ (∃𝑥𝐴 𝜑 ∨ ∃𝑥𝐴 𝜓)))
41, 3bitr4id 199 1 (∃𝑦 𝑦𝐴 → (∃𝑥𝐴 (𝜑𝜓) ↔ (∃𝑥𝐴 𝜑𝜓)))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 105  wo 709  wex 1503  wcel 2160  wrex 2469
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-ext 2171
This theorem depends on definitions:  df-bi 117  df-cleq 2182  df-clel 2185  df-rex 2474
This theorem is referenced by:  frecabcl  6424
  Copyright terms: Public domain W3C validator