Theorem bj-bi3ant 37439

Description: This used to be in the main part. (Contributed by Wolf Lammen, 14-May-2013.) (Revised by BJ, 14-Jun-2019.)

Hypothesis

Ref	Expression
bj-bi3ant.1	⊢ (𝜑 → (𝜓 → 𝜒))

Assertion

Ref	Expression
bj-bi3ant	⊢ (((𝜃 → 𝜏) → 𝜑) → (((𝜏 → 𝜃) → 𝜓) → ((𝜃 ↔ 𝜏) → 𝜒)))

Proof of Theorem bj-bi3ant

Step	Hyp	Ref	Expression
1		biimp 218	. . 3 ⊢ ((𝜃 ↔ 𝜏) → (𝜃 → 𝜏))
2	1	imim1i 64	. 2 ⊢ (((𝜃 → 𝜏) → 𝜑) → ((𝜃 ↔ 𝜏) → 𝜑))
3		biimpr 223	. . 3 ⊢ ((𝜃 ↔ 𝜏) → (𝜏 → 𝜃))
4	3	imim1i 64	. 2 ⊢ (((𝜏 → 𝜃) → 𝜓) → ((𝜃 ↔ 𝜏) → 𝜓))
5		bj-bi3ant.1	. . 3 ⊢ (𝜑 → (𝜓 → 𝜒))
6	5	imim3i 65	. 2 ⊢ (((𝜃 ↔ 𝜏) → 𝜑) → (((𝜃 ↔ 𝜏) → 𝜓) → ((𝜃 ↔ 𝜏) → 𝜒)))
7	2, 4, 6	syl2im 41	1 ⊢ (((𝜃 → 𝜏) → 𝜑) → (((𝜏 → 𝜃) → 𝜓) → ((𝜃 ↔ 𝜏) → 𝜒)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209

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

This theorem is used by:  bj-bisym  37440