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Metamath Proof Explorer

Theorem 3impexpbicomiVD

Description: Virtual deduction proof of 3impexpbicomi . The following user's proof is completed by invoking mmj2's unify command and using mmj2's StepSelector to pick all remaining steps of the Metamath proof.

h1::	\|- ( ( ph /\ ps /\ ch ) -> ( th <-> ta ) )
qed:1,?: e0a	\|- ( ph -> ( ps -> ( ch -> ( ta <-> th ) ) ) )

(Contributed by Alan Sare, 31-Dec-2011) (Proof modification is discouraged.) (New usage is discouraged.)

		Ref	Expression
	Hypothesis	3impexpbicomiVD.1	⊢ ( ( 𝜑 ∧ 𝜓 ∧ 𝜒 ) → ( 𝜃 ↔ 𝜏 ) )
	Assertion	3impexpbicomiVD	⊢ ( 𝜑 → ( 𝜓 → ( 𝜒 → ( 𝜏 ↔ 𝜃 ) ) ) )

Proof

Step	Hyp	Ref	Expression
1		3impexpbicomiVD.1	⊢ ( ( 𝜑 ∧ 𝜓 ∧ 𝜒 ) → ( 𝜃 ↔ 𝜏 ) )
2		3impexpbicom	⊢ ( ( ( 𝜑 ∧ 𝜓 ∧ 𝜒 ) → ( 𝜃 ↔ 𝜏 ) ) ↔ ( 𝜑 → ( 𝜓 → ( 𝜒 → ( 𝜏 ↔ 𝜃 ) ) ) ) )
3	2	biimpi	⊢ ( ( ( 𝜑 ∧ 𝜓 ∧ 𝜒 ) → ( 𝜃 ↔ 𝜏 ) ) → ( 𝜑 → ( 𝜓 → ( 𝜒 → ( 𝜏 ↔ 𝜃 ) ) ) ) )
4	1 3	e0a	⊢ ( 𝜑 → ( 𝜓 → ( 𝜒 → ( 𝜏 ↔ 𝜃 ) ) ) )