Theorem expandan

Description: Expand conjunction to primitives. (Contributed by Rohan Ridenour, 13-Aug-2023)

	Ref	Expression
Hypotheses	expandan.1	⊢ ( 𝜑 ↔ 𝜓 )
	expandan.2	⊢ ( 𝜒 ↔ 𝜃 )
Assertion	expandan	⊢ ( ( 𝜑 ∧ 𝜒 ) ↔ ¬ ( 𝜓 → ¬ 𝜃 ) )

Proof

Step	Hyp	Ref	Expression
1		expandan.1	⊢ ( 𝜑 ↔ 𝜓 )
2		expandan.2	⊢ ( 𝜒 ↔ 𝜃 )
3	1 2	anbi12i	⊢ ( ( 𝜑 ∧ 𝜒 ) ↔ ( 𝜓 ∧ 𝜃 ) )
4		df-an	⊢ ( ( 𝜓 ∧ 𝜃 ) ↔ ¬ ( 𝜓 → ¬ 𝜃 ) )
5	3 4	bitri	⊢ ( ( 𝜑 ∧ 𝜒 ) ↔ ¬ ( 𝜓 → ¬ 𝜃 ) )