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Metamath Proof Explorer

Theorem ee11an

Description: e11an without virtual deductions. syl22anc is also e11an without virtual deductions, exept with a different order of hypotheses. (Contributed by Alan Sare, 8-Jul-2011) (Proof modification is discouraged.) (New usage is discouraged.)

	Ref	Expression
Hypotheses	ee11an.1	⊢ ( 𝜑 → 𝜓 )
	ee11an.2	⊢ ( 𝜑 → 𝜒 )
	ee11an.3	⊢ ( ( 𝜓 ∧ 𝜒 ) → 𝜃 )
Assertion	ee11an	⊢ ( 𝜑 → 𝜃 )

Proof

Step	Hyp	Ref	Expression
1		ee11an.1	⊢ ( 𝜑 → 𝜓 )
2		ee11an.2	⊢ ( 𝜑 → 𝜒 )
3		ee11an.3	⊢ ( ( 𝜓 ∧ 𝜒 ) → 𝜃 )
4	3	ex	⊢ ( 𝜓 → ( 𝜒 → 𝜃 ) )
5	1 2 4	sylc	⊢ ( 𝜑 → 𝜃 )