MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  nanbi2 Structured version   Visualization version   GIF version

Theorem nanbi2 1453
Description: Introduce a left anti-conjunct to both sides of a logical equivalence. (Contributed by SF, 2-Jan-2018.)
Assertion
Ref Expression
nanbi2 ((𝜑𝜓) → ((𝜒𝜑) ↔ (𝜒𝜓)))

Proof of Theorem nanbi2
StepHypRef Expression
1 nanbi1 1452 . 2 ((𝜑𝜓) → ((𝜑𝜒) ↔ (𝜓𝜒)))
2 nancom 1447 . 2 ((𝜒𝜑) ↔ (𝜑𝜒))
3 nancom 1447 . 2 ((𝜒𝜓) ↔ (𝜓𝜒))
41, 2, 33bitr4g 303 1 ((𝜑𝜓) → ((𝜒𝜑) ↔ (𝜒𝜓)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wnan 1444
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 386  df-nan 1445
This theorem is referenced by:  nanbi12  1454  nanbi2i  1456  nanbi2d  1459  nabi2  32085  rp-fakenanass  37380
  Copyright terms: Public domain W3C validator