Intuitionistic Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  ILE Home  >  Th. List  >  19.9hd GIF version

Theorem 19.9hd 1638
 Description: A deduction version of one direction of 19.9 1620. This is an older variation of this theorem; new proofs should use 19.9d 1637. (Contributed by NM, 5-Aug-1993.) (New usage is discouraged.)
Hypotheses
Ref Expression
19.9hd.1 (𝜓 → ∀𝑥𝜓)
19.9hd.2 (𝜓 → (𝜑 → ∀𝑥𝜑))
Assertion
Ref Expression
19.9hd (𝜓 → (∃𝑥𝜑𝜑))

Proof of Theorem 19.9hd
StepHypRef Expression
1 19.9hd.1 . 2 (𝜓 → ∀𝑥𝜓)
2 19.9hd.2 . . 3 (𝜓 → (𝜑 → ∀𝑥𝜑))
32alimi 1432 . 2 (∀𝑥𝜓 → ∀𝑥(𝜑 → ∀𝑥𝜑))
4 19.9ht 1617 . 2 (∀𝑥(𝜑 → ∀𝑥𝜑) → (∃𝑥𝜑𝜑))
51, 3, 43syl 17 1 (𝜓 → (∃𝑥𝜑𝜑))
 Colors of variables: wff set class Syntax hints:   → wi 4  ∀wal 1330  ∃wex 1469 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-5 1424  ax-gen 1426  ax-ie2 1471 This theorem depends on definitions:  df-bi 116 This theorem is referenced by:  sbiedh  1756  bj-sbimedh  13188
 Copyright terms: Public domain W3C validator