Theorem pm5.21nd

Description: Eliminate an antecedent implied by each side of a biconditional. Variant of pm5.21ndd . (Contributed by NM, 20-Nov-2005) (Proof shortened by Wolf Lammen, 4-Nov-2013)

	Ref	Expression
Hypotheses	pm5.21nd.1	⊢ ( ( 𝜑 ∧ 𝜓 ) → 𝜃 )
	pm5.21nd.2	⊢ ( ( 𝜑 ∧ 𝜒 ) → 𝜃 )
	pm5.21nd.3	⊢ ( 𝜃 → ( 𝜓 ↔ 𝜒 ) )
Assertion	pm5.21nd	⊢ ( 𝜑 → ( 𝜓 ↔ 𝜒 ) )

Proof

Step	Hyp	Ref	Expression
1		pm5.21nd.1	⊢ ( ( 𝜑 ∧ 𝜓 ) → 𝜃 )
2		pm5.21nd.2	⊢ ( ( 𝜑 ∧ 𝜒 ) → 𝜃 )
3		pm5.21nd.3	⊢ ( 𝜃 → ( 𝜓 ↔ 𝜒 ) )
4	1	ex	⊢ ( 𝜑 → ( 𝜓 → 𝜃 ) )
5	2	ex	⊢ ( 𝜑 → ( 𝜒 → 𝜃 ) )
6	3	a1i	⊢ ( 𝜑 → ( 𝜃 → ( 𝜓 ↔ 𝜒 ) ) )
7	4 5 6	pm5.21ndd	⊢ ( 𝜑 → ( 𝜓 ↔ 𝜒 ) )