ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  19.23h GIF version

Theorem 19.23h 1428
Description: Theorem 19.23 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 1-Feb-2015.)
Hypothesis
Ref Expression
19.23h.1 (𝜓 → ∀𝑥𝜓)
Assertion
Ref Expression
19.23h (∀𝑥(𝜑𝜓) ↔ (∃𝑥𝜑𝜓))

Proof of Theorem 19.23h
StepHypRef Expression
1 19.23h.1 . . 3 (𝜓 → ∀𝑥𝜓)
21ax-gen 1379 . 2 𝑥(𝜓 → ∀𝑥𝜓)
3 19.23ht 1427 . 2 (∀𝑥(𝜓 → ∀𝑥𝜓) → (∀𝑥(𝜑𝜓) ↔ (∃𝑥𝜑𝜓)))
42, 3ax-mp 7 1 (∀𝑥(𝜑𝜓) ↔ (∃𝑥𝜑𝜓))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 103  wal 1283  wex 1422
This theorem was proved from axioms:  ax-mp 7  ax-gen 1379  ax-ie2 1424
This theorem is referenced by:  alnex  1429  19.8a  1523  exlimih  1525  exlimdh  1528  nf2  1599  equs5or  1753  19.23v  1806  pm11.53  1818
  Copyright terms: Public domain W3C validator