Theorem expandan 45056

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

Hypotheses

Ref	Expression
expandan.1	⊢ (𝜑 ↔ 𝜓)
expandan.2	⊢ (𝜒 ↔ 𝜃)

Assertion

Ref	Expression
expandan	⊢ ((𝜑 ∧ 𝜒) ↔ ¬ (𝜓 → ¬ 𝜃))

Proof of Theorem expandan

Step	Hyp	Ref	Expression
1		expandan.1	. . 3 ⊢ (𝜑 ↔ 𝜓)
2		expandan.2	. . 3 ⊢ (𝜒 ↔ 𝜃)
3	1, 2	anbi12i 640	. 2 ⊢ ((𝜑 ∧ 𝜒) ↔ (𝜓 ∧ 𝜃))
4		df-an 402	. 2 ⊢ ((𝜓 ∧ 𝜃) ↔ ¬ (𝜓 → ¬ 𝜃))
5	3, 4	bitri 278	1 ⊢ ((𝜑 ∧ 𝜒) ↔ ¬ (𝜓 → ¬ 𝜃))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This proof depends on definitions:  df-bi 210  df-an 402

This theorem is used by:  ismnuprim  45062  rr-grothprimbi  45063