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Theorem expandan 40779
Description: Expand conjunction to primitives. (Contributed by Rohan Ridenour, 13-Aug-2023.)
Hypotheses
Ref Expression
expandan.1 (𝜑𝜓)
expandan.2 (𝜒𝜃)
Assertion
Ref Expression
expandan ((𝜑𝜒) ↔ ¬ (𝜓 → ¬ 𝜃))

Proof of Theorem expandan
StepHypRef Expression
1 expandan.1 . . 3 (𝜑𝜓)
2 expandan.2 . . 3 (𝜒𝜃)
31, 2anbi12i 628 . 2 ((𝜑𝜒) ↔ (𝜓𝜃))
4 df-an 399 . 2 ((𝜓𝜃) ↔ ¬ (𝜓 → ¬ 𝜃))
53, 4bitri 277 1 ((𝜑𝜒) ↔ ¬ (𝜓 → ¬ 𝜃))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 208  wa 398
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 209  df-an 399
This theorem is referenced by:  ismnuprim  40785  rr-grothprimbi  40786
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