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

Theorem 19.28 1543
Description: Theorem 19.28 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)
Hypothesis
Ref Expression
19.28.1 𝑥𝜑
Assertion
Ref Expression
19.28 (∀𝑥(𝜑𝜓) ↔ (𝜑 ∧ ∀𝑥𝜓))

Proof of Theorem 19.28
StepHypRef Expression
1 19.26 1458 . 2 (∀𝑥(𝜑𝜓) ↔ (∀𝑥𝜑 ∧ ∀𝑥𝜓))
2 19.28.1 . . . 4 𝑥𝜑
3219.3 1534 . . 3 (∀𝑥𝜑𝜑)
43anbi1i 454 . 2 ((∀𝑥𝜑 ∧ ∀𝑥𝜓) ↔ (𝜑 ∧ ∀𝑥𝜓))
51, 4bitri 183 1 (∀𝑥(𝜑𝜓) ↔ (𝜑 ∧ ∀𝑥𝜓))
Colors of variables: wff set class
Syntax hints:  wa 103  wb 104  wal 1330  wnf 1437
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 1424  ax-gen 1426  ax-4 1488
This theorem depends on definitions:  df-bi 116  df-nf 1438
This theorem is referenced by:  aaan  1567
  Copyright terms: Public domain W3C validator