Theorem bi3ant 280

Description: Construct a bi-conditional 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 178	. . 3 ⊢ ((θ ↔ τ) → (θ → τ))
2	1	imim1i 54	. 2 ⊢ (((θ → τ) → φ) → ((θ ↔ τ) → φ))
3		bi2 189	. . 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 176

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 177

This theorem is referenced by:  bisym  281