ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  19.28h GIF version

Theorem 19.28h 1497
Description: Theorem 19.28 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)
Hypothesis
Ref Expression
19.28h.1 (𝜑 → ∀𝑥𝜑)
Assertion
Ref Expression
19.28h (∀𝑥(𝜑𝜓) ↔ (𝜑 ∧ ∀𝑥𝜓))

Proof of Theorem 19.28h
StepHypRef Expression
1 19.26 1413 . 2 (∀𝑥(𝜑𝜓) ↔ (∀𝑥𝜑 ∧ ∀𝑥𝜓))
2 19.28h.1 . . . 4 (𝜑 → ∀𝑥𝜑)
3219.3h 1488 . . 3 (∀𝑥𝜑𝜑)
43anbi1i 446 . 2 ((∀𝑥𝜑 ∧ ∀𝑥𝜓) ↔ (𝜑 ∧ ∀𝑥𝜓))
51, 4bitri 182 1 (∀𝑥(𝜑𝜓) ↔ (𝜑 ∧ ∀𝑥𝜓))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wb 103  wal 1285
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 1379  ax-gen 1381  ax-4 1443
This theorem depends on definitions:  df-bi 115
This theorem is referenced by:  nfan1  1499  aaanh  1521  exan  1626  19.28v  1825
  Copyright terms: Public domain W3C validator