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Metamath Proof Explorer

Theorem impcon4bid

Description: A variation on impbid with contraposition. (Contributed by Jeff Hankins, 3-Jul-2009)

Ref Expression
Hypotheses impcon4bid.1 ⊢ ( 𝜑 → ( 𝜓 → 𝜒 ) )
impcon4bid.2 ⊢ ( 𝜑 → ( ¬ 𝜓 → ¬ 𝜒 ) )
Assertion impcon4bid ( 𝜑 → ( 𝜓 ↔ 𝜒 ) )

Proof

Step Hyp Ref Expression
1 impcon4bid.1 ⊢ ( 𝜑 → ( 𝜓 → 𝜒 ) )
2 impcon4bid.2 ⊢ ( 𝜑 → ( ¬ 𝜓 → ¬ 𝜒 ) )
3 2 con4d ⊢ ( 𝜑 → ( 𝜒 → 𝜓 ) )
4 1 3 impbid ⊢ ( 𝜑 → ( 𝜓 ↔ 𝜒 ) )