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

Theorem rmobida 2617
 Description: Formula-building rule for restricted existential quantifier (deduction form). (Contributed by NM, 16-Jun-2017.)
Hypotheses
Ref Expression
rmobida.1 𝑥𝜑
rmobida.2 ((𝜑𝑥𝐴) → (𝜓𝜒))
Assertion
Ref Expression
rmobida (𝜑 → (∃*𝑥𝐴 𝜓 ↔ ∃*𝑥𝐴 𝜒))

Proof of Theorem rmobida
StepHypRef Expression
1 rmobida.1 . . 3 𝑥𝜑
2 rmobida.2 . . . 4 ((𝜑𝑥𝐴) → (𝜓𝜒))
32pm5.32da 447 . . 3 (𝜑 → ((𝑥𝐴𝜓) ↔ (𝑥𝐴𝜒)))
41, 3mobid 2034 . 2 (𝜑 → (∃*𝑥(𝑥𝐴𝜓) ↔ ∃*𝑥(𝑥𝐴𝜒)))
5 df-rmo 2424 . 2 (∃*𝑥𝐴 𝜓 ↔ ∃*𝑥(𝑥𝐴𝜓))
6 df-rmo 2424 . 2 (∃*𝑥𝐴 𝜒 ↔ ∃*𝑥(𝑥𝐴𝜒))
74, 5, 63bitr4g 222 1 (𝜑 → (∃*𝑥𝐴 𝜓 ↔ ∃*𝑥𝐴 𝜒))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104  Ⅎwnf 1436   ∈ wcel 1480  ∃*wmo 2000  ∃*wrmo 2419 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 1423  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-4 1487  ax-17 1506  ax-ial 1514 This theorem depends on definitions:  df-bi 116  df-nf 1437  df-eu 2002  df-mo 2003  df-rmo 2424 This theorem is referenced by:  rmobidva  2618
 Copyright terms: Public domain W3C validator