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Metamath Proof Explorer

Theorem ee123

Description: e123 without virtual deductions. (Contributed by Alan Sare, 25-Jul-2011) (Proof modification is discouraged.) (New usage is discouraged.)

	Ref	Expression
Hypotheses	ee123.1	$⊢ φ \to ψ$
	ee123.2	$⊢ φ \to (χ \to θ)$
	ee123.3	$⊢ φ \to (χ \to (τ \to η))$
	ee123.4	$⊢ ψ \to (θ \to (η \to ζ))$
Assertion	ee123	$⊢ φ \to (χ \to (τ \to ζ))$

Proof

Step	Hyp	Ref	Expression
1		ee123.1	$⊢ φ \to ψ$
2		ee123.2	$⊢ φ \to (χ \to θ)$
3		ee123.3	$⊢ φ \to (χ \to (τ \to η))$
4		ee123.4	$⊢ ψ \to (θ \to (η \to ζ))$
5	1	a1d	$⊢ φ \to (τ \to ψ)$
6	5	a1d	$⊢ φ \to (χ \to (τ \to ψ))$
7	2	a1dd	$⊢ φ \to (χ \to (τ \to θ))$
8	6 7 3 4	ee333	$⊢ φ \to (χ \to (τ \to ζ))$