Theorem bi3ant 213

Description: Construct a biconditional in antecedent position. (Contributed by Wolf Lammen, 14-May-2013.)

Hypothesis

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

Assertion

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

Proof of Theorem bi3ant

Step	Hyp	Ref	Expression
1		bi1 111	. . 3 ⊢ ((𝜃 ↔ 𝜏) → (𝜃 → 𝜏))
2	1	imim1i 54	. 2 ⊢ (((𝜃 → 𝜏) → 𝜑) → ((𝜃 ↔ 𝜏) → 𝜑))
3		bi2 121	. . 3 ⊢ ((𝜃 ↔ 𝜏) → (𝜏 → 𝜃))
4	3	imim1i 54	. 2 ⊢ (((𝜏 → 𝜃) → 𝜓) → ((𝜃 ↔ 𝜏) → 𝜓))
5		bi3ant.1	. . 3 ⊢ (𝜑 → (𝜓 → 𝜒))
6	5	imim3i 55	. 2 ⊢ (((𝜃 ↔ 𝜏) → 𝜑) → (((𝜃 ↔ 𝜏) → 𝜓) → ((𝜃 ↔ 𝜏) → 𝜒)))
7	2, 4, 6	syl2im 34	1 ⊢ (((𝜃 → 𝜏) → 𝜑) → (((𝜏 → 𝜃) → 𝜓) → ((𝜃 ↔ 𝜏) → 𝜒)))

Syntax hints:   → wi 4   ↔ wb 98

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100

This theorem depends on definitions:  df-bi 110

This theorem is referenced by:  bisym  214