Theorem eelTT1

Description: An elimination deduction. (Contributed by Alan Sare, 4-Feb-2017) (Proof modification is discouraged.) (New usage is discouraged.)

	Ref	Expression
Hypotheses	eelTT1.1	⊢ ( ⊤ → 𝜑 )
	eelTT1.2	⊢ ( ⊤ → 𝜓 )
	eelTT1.3	⊢ ( 𝜒 → 𝜃 )
	eelTT1.4	⊢ ( ( 𝜑 ∧ 𝜓 ∧ 𝜃 ) → 𝜏 )
Assertion	eelTT1	⊢ ( 𝜒 → 𝜏 )

Proof

Step	Hyp	Ref	Expression
1		eelTT1.1	⊢ ( ⊤ → 𝜑 )
2		eelTT1.2	⊢ ( ⊤ → 𝜓 )
3		eelTT1.3	⊢ ( 𝜒 → 𝜃 )
4		eelTT1.4	⊢ ( ( 𝜑 ∧ 𝜓 ∧ 𝜃 ) → 𝜏 )
5		3anass	⊢ ( ( ⊤ ∧ ⊤ ∧ 𝜒 ) ↔ ( ⊤ ∧ ( ⊤ ∧ 𝜒 ) ) )
6		anabs5	⊢ ( ( ⊤ ∧ ( ⊤ ∧ 𝜒 ) ) ↔ ( ⊤ ∧ 𝜒 ) )
7		truan	⊢ ( ( ⊤ ∧ 𝜒 ) ↔ 𝜒 )
8	5 6 7	3bitri	⊢ ( ( ⊤ ∧ ⊤ ∧ 𝜒 ) ↔ 𝜒 )
9	1 4	syl3an1	⊢ ( ( ⊤ ∧ 𝜓 ∧ 𝜃 ) → 𝜏 )
10	2 9	syl3an2	⊢ ( ( ⊤ ∧ ⊤ ∧ 𝜃 ) → 𝜏 )
11	3 10	syl3an3	⊢ ( ( ⊤ ∧ ⊤ ∧ 𝜒 ) → 𝜏 )
12	8 11	sylbir	⊢ ( 𝜒 → 𝜏 )