Theorem con4bid 284

Description: A contraposition deduction. (Contributed by NM, 21-May-1994.)

Hypothesis

Ref	Expression
con4bid.1	⊢ (φ → (¬ ψ ↔ ¬ χ))

Assertion

Ref	Expression
con4bid	⊢ (φ → (ψ ↔ χ))

Proof of Theorem con4bid

Step	Hyp	Ref	Expression
1		con4bid.1	. . . 4 ⊢ (φ → (¬ ψ ↔ ¬ χ))
2	1	biimprd 214	. . 3 ⊢ (φ → (¬ χ → ¬ ψ))
3	2	con4d 97	. 2 ⊢ (φ → (ψ → χ))
4	1	biimpd 198	. 2 ⊢ (φ → (¬ ψ → ¬ χ))
5	3, 4	impcon4bid 196	1 ⊢ (φ → (ψ ↔ χ))

Syntax hints:  ¬ wn 3   → 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:  notbid  285  notbi  286  had0  1403  necon4abid  2580  sbcne12g  3154  isomin  5496