Theorem pm5.21ndd

Description: Eliminate an antecedent implied by each side of a biconditional, deduction version. (Contributed by Paul Chapman, 21-Nov-2012) (Proof shortened by Wolf Lammen, 6-Oct-2013)

	Ref	Expression
Hypotheses	pm5.21ndd.1	$$⊢\phi \to \left(\chi \to \psi \right)$$
	pm5.21ndd.2	$$⊢\phi \to \left(\theta \to \psi \right)$$
	pm5.21ndd.3	$$⊢\phi \to \left(\psi \to \left(\chi \leftrightarrow \theta \right)\right)$$
Assertion	pm5.21ndd	$$⊢\phi \to \left(\chi \leftrightarrow \theta \right)$$

Proof

Step	Hyp	Ref	Expression
1		pm5.21ndd.1	$$⊢\phi \to \left(\chi \to \psi \right)$$
2		pm5.21ndd.2	$$⊢\phi \to \left(\theta \to \psi \right)$$
3		pm5.21ndd.3	$$⊢\phi \to \left(\psi \to \left(\chi \leftrightarrow \theta \right)\right)$$
4	1	con3d	$$⊢\phi \to \left(\neg \psi \to \neg \chi \right)$$
5	2	con3d	$$⊢\phi \to \left(\neg \psi \to \neg \theta \right)$$
6		pm5.21im	$$⊢\neg \chi \to \left(\neg \theta \to \left(\chi \leftrightarrow \theta \right)\right)$$
7	4 5 6	syl6c	$$⊢\phi \to \left(\neg \psi \to \left(\chi \leftrightarrow \theta \right)\right)$$
8	3 7	pm2.61d	$$⊢\phi \to \left(\chi \leftrightarrow \theta \right)$$