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Metamath Proof Explorer

Theorem impbid

Description: Deduce an equivalence from two implications. Deduction associated with impbi and impbii . (Contributed by NM, 24-Jan-1993) Prove it from impbid21d . (Revised by Wolf Lammen, 3-Nov-2012)

	Ref	Expression
Hypotheses	impbid.1	⊢ ( 𝜑 → ( 𝜓 → 𝜒 ) )
	impbid.2	⊢ ( 𝜑 → ( 𝜒 → 𝜓 ) )
Assertion	impbid	⊢ ( 𝜑 → ( 𝜓 ↔ 𝜒 ) )

Proof

Step	Hyp	Ref	Expression
1		impbid.1	⊢ ( 𝜑 → ( 𝜓 → 𝜒 ) )
2		impbid.2	⊢ ( 𝜑 → ( 𝜒 → 𝜓 ) )
3	1 2	impbid21d	⊢ ( 𝜑 → ( 𝜑 → ( 𝜓 ↔ 𝜒 ) ) )
4	3	pm2.43i	⊢ ( 𝜑 → ( 𝜓 ↔ 𝜒 ) )