Metamath Proof Explorer

Theorem ee333

Description: e333 without virtual deductions. (Contributed by Alan Sare, 17-Jul-2011) (Proof modification is discouraged.) (New usage is discouraged.)

		Ref	Expression
	Hypotheses	ee333.1	$⊢ φ \to (ψ \to (χ \to θ))$
		ee333.2	$⊢ φ \to (ψ \to (χ \to τ))$
		ee333.3	$⊢ φ \to (ψ \to (χ \to η))$
		ee333.4	$⊢ θ \to (τ \to (η \to ζ))$
	Assertion	ee333	$⊢ φ \to (ψ \to (χ \to ζ))$

Proof

Step	Hyp	Ref	Expression
1		ee333.1	$⊢ φ \to (ψ \to (χ \to θ))$
2		ee333.2	$⊢ φ \to (ψ \to (χ \to τ))$
3		ee333.3	$⊢ φ \to (ψ \to (χ \to η))$
4		ee333.4	$⊢ θ \to (τ \to (η \to ζ))$
5	1	3imp	$⊢ (φ \land ψ \land χ) \to θ$
6	2	3imp	$⊢ (φ \land ψ \land χ) \to τ$
7	3	3imp	$⊢ (φ \land ψ \land χ) \to η$
8	5 6 7 4	syl3c	$⊢ (φ \land ψ \land χ) \to ζ$
9	8	3exp	$⊢ φ \to (ψ \to (χ \to ζ))$