 Mathbox for Alan Sare < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  ancomstVD Structured version   Visualization version   GIF version

Theorem ancomstVD 39415
Description: Closed form of ancoms 468. The following user's proof is completed by invoking mmj2's unify command and using mmj2's StepSelector to pick all remaining steps of the Metamath proof.
 1:: ⊢ ((𝜑 ∧ 𝜓) ↔ (𝜓 ∧ 𝜑)) qed:1,?: e0a 39316 ⊢ (((𝜑 ∧ 𝜓) → 𝜒) ↔ ((𝜓 ∧ 𝜑) → 𝜒))
The proof of ancomst 467 is derived automatically from it. (Contributed by Alan Sare, 25-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
ancomstVD (((𝜑𝜓) → 𝜒) ↔ ((𝜓𝜑) → 𝜒))

Proof of Theorem ancomstVD
StepHypRef Expression
1 ancom 465 . 2 ((𝜑𝜓) ↔ (𝜓𝜑))
2 imbi1 336 . 2 (((𝜑𝜓) ↔ (𝜓𝜑)) → (((𝜑𝜓) → 𝜒) ↔ ((𝜓𝜑) → 𝜒)))
31, 2e0a 39316 1 (((𝜑𝜓) → 𝜒) ↔ ((𝜓𝜑) → 𝜒))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 383 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 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator