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Theorem nanbi2 1486
Description: Introduce a left anti-conjunct to both sides of a logical equivalence. (Contributed by Anthony Hart, 1-Sep-2011.) (Proof shortened by SF, 2-Jan-2018.)
Assertion
Ref Expression
nanbi2 ((𝜑𝜓) → ((𝜒𝜑) ↔ (𝜒𝜓)))

Proof of Theorem nanbi2
StepHypRef Expression
1 nanbi1 1485 . 2 ((𝜑𝜓) → ((𝜑𝜒) ↔ (𝜓𝜒)))
2 nancom 1480 . 2 ((𝜒𝜑) ↔ (𝜑𝜒))
3 nancom 1480 . 2 ((𝜒𝜓) ↔ (𝜓𝜒))
41, 2, 33bitr4g 315 1 ((𝜑𝜓) → ((𝜒𝜑) ↔ (𝜒𝜓)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wnan 1475
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 208  df-an 397  df-nan 1476
This theorem is referenced by:  nanbi12  1487  nanbi2i  1489  nanbi2d  1492  nanass  1494
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