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Theorem nanbi1 1295
Description: Introduce a right anti-conjunct to both sides of a logical equivalence. (Contributed by SF, 2-Jan-2018.)
Assertion
Ref Expression
nanbi1 ((φψ) → ((φ χ) ↔ (ψ χ)))

Proof of Theorem nanbi1
StepHypRef Expression
1 anbi1 687 . . 3 ((φψ) → ((φ χ) ↔ (ψ χ)))
21notbid 285 . 2 ((φψ) → (¬ (φ χ) ↔ ¬ (ψ χ)))
3 df-nan 1288 . 2 ((φ χ) ↔ ¬ (φ χ))
4 df-nan 1288 . 2 ((ψ χ) ↔ ¬ (ψ χ))
52, 3, 43bitr4g 279 1 ((φψ) → ((φ χ) ↔ (ψ χ)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 176   wa 358   wnan 1287
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 177  df-an 360  df-nan 1288
This theorem is referenced by:  nanbi2  1296  nanbi12  1297  nanbi1i  1298  nanbi1d  1301
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