Theorem resinsnlem 49529

Description: Lemma for resinsnALT 49531. (Contributed by Zhi Wang, 6-Oct-2025.)

Hypotheses

Ref	Expression
resinsnlem.1	⊢ (𝜑 → (𝜒 ↔ ¬ 𝜓))
resinsnlem.2	⊢ (¬ 𝜑 → 𝜒)

Assertion

Ref	Expression
resinsnlem	⊢ ((𝜑 ∧ 𝜓) ↔ ¬ 𝜒)

Proof of Theorem resinsnlem

Step	Hyp	Ref	Expression
1		resinsnlem.1	. . . 4 ⊢ (𝜑 → (𝜒 ↔ ¬ 𝜓))
2	1	con2bid 357	. . 3 ⊢ (𝜑 → (𝜓 ↔ ¬ 𝜒))
3	2	biimpa 481	. 2 ⊢ ((𝜑 ∧ 𝜓) → ¬ 𝜒)
4		resinsnlem.2	. . . 4 ⊢ (¬ 𝜑 → 𝜒)
5	4	con1i 148	. . 3 ⊢ (¬ 𝜒 → 𝜑)
6	5, 2	syl 18	. . . 4 ⊢ (¬ 𝜒 → (𝜓 ↔ ¬ 𝜒))
7	6	ibir 271	. . 3 ⊢ (¬ 𝜒 → 𝜓)
8	5, 7	jca 520	. 2 ⊢ (¬ 𝜒 → (𝜑 ∧ 𝜓))
9	3, 8	impbii 212	1 ⊢ ((𝜑 ∧ 𝜓) ↔ ¬ 𝜒)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 400

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 210  df-an 401

This theorem is referenced by:  resinsnALT  49531