Theorem wl-nanbi1 32479

Description: Introduce a right anti-conjunct to both sides of a logical equivalence. (Contributed by SF, 2-Jan-2018.) (Revised by Wolf Lammen, 27-Jun-2020.)

Assertion

Ref	Expression
wl-nanbi1	⊢ ((𝜑 ↔ 𝜓) → ((𝜑 ⊼ 𝜒) ↔ (𝜓 ⊼ 𝜒)))

Proof of Theorem wl-nanbi1

Step	Hyp	Ref	Expression
1		imbi1 336	. 2 ⊢ ((𝜑 ↔ 𝜓) → ((𝜑 → ¬ 𝜒) ↔ (𝜓 → ¬ 𝜒)))
2		wl-dfnan2 32475	. 2 ⊢ ((𝜑 ⊼ 𝜒) ↔ (𝜑 → ¬ 𝜒))
3		wl-dfnan2 32475	. 2 ⊢ ((𝜓 ⊼ 𝜒) ↔ (𝜓 → ¬ 𝜒))
4	1, 2, 3	3bitr4g 302	1 ⊢ ((𝜑 ↔ 𝜓) → ((𝜑 ⊼ 𝜒) ↔ (𝜓 ⊼ 𝜒)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ⊼ wnan 1439

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 196  df-an 385  df-nan 1440

This theorem is referenced by: (None)