Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  nic-isw2 Structured version   Visualization version   GIF version

Theorem nic-isw2 1755
 Description: Inference for swapping nested terms. (Contributed by Jeff Hoffman, 17-Nov-2007.) (Proof modification is discouraged.) (New usage is discouraged.)
Hypothesis
Ref Expression
nic-isw2.1 (𝜓 ⊼ (𝜃𝜑))
Assertion
Ref Expression
nic-isw2 (𝜓 ⊼ (𝜑𝜃))

Proof of Theorem nic-isw2
StepHypRef Expression
1 nic-isw2.1 . . 3 (𝜓 ⊼ (𝜃𝜑))
2 nic-swap 1753 . . . 4 ((𝜑𝜃) ⊼ ((𝜃𝜑) ⊼ (𝜃𝜑)))
32nic-imp 1749 . . 3 ((𝜓 ⊼ (𝜃𝜑)) ⊼ (((𝜑𝜃) ⊼ 𝜓) ⊼ ((𝜑𝜃) ⊼ 𝜓)))
41, 3nic-mp 1745 . 2 ((𝜑𝜃) ⊼ 𝜓)
54nic-isw1 1754 1 (𝜓 ⊼ (𝜑𝜃))
 Colors of variables: wff setvar class Syntax hints:   ⊼ wnan 1596 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8 This theorem depends on definitions:  df-bi 197  df-an 385  df-nan 1597 This theorem is referenced by:  nic-bi1  1762  nic-bi2  1763  nic-luk1  1765  nic-luk2  1766
 Copyright terms: Public domain W3C validator