Theorem eelT01

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

	Ref	Expression
Hypotheses	eelT01.1	⊢ ( ⊤ → 𝜑 )
	eelT01.2	⊢ 𝜓
	eelT01.3	⊢ ( 𝜒 → 𝜃 )
	eelT01.4	⊢ ( ( 𝜑 ∧ 𝜓 ∧ 𝜃 ) → 𝜏 )
Assertion	eelT01	⊢ ( 𝜒 → 𝜏 )

Proof

Step	Hyp	Ref	Expression
1		eelT01.1	⊢ ( ⊤ → 𝜑 )
2		eelT01.2	⊢ 𝜓
3		eelT01.3	⊢ ( 𝜒 → 𝜃 )
4		eelT01.4	⊢ ( ( 𝜑 ∧ 𝜓 ∧ 𝜃 ) → 𝜏 )
5		3anass	⊢ ( ( ⊤ ∧ 𝜓 ∧ 𝜒 ) ↔ ( ⊤ ∧ ( 𝜓 ∧ 𝜒 ) ) )
6		truan	⊢ ( ( ⊤ ∧ ( 𝜓 ∧ 𝜒 ) ) ↔ ( 𝜓 ∧ 𝜒 ) )
7		simpr	⊢ ( ( 𝜓 ∧ 𝜒 ) → 𝜒 )
8	2	jctl	⊢ ( 𝜒 → ( 𝜓 ∧ 𝜒 ) )
9	7 8	impbii	⊢ ( ( 𝜓 ∧ 𝜒 ) ↔ 𝜒 )
10	5 6 9	3bitri	⊢ ( ( ⊤ ∧ 𝜓 ∧ 𝜒 ) ↔ 𝜒 )
11	1 4	syl3an1	⊢ ( ( ⊤ ∧ 𝜓 ∧ 𝜃 ) → 𝜏 )
12	3 11	syl3an3	⊢ ( ( ⊤ ∧ 𝜓 ∧ 𝜒 ) → 𝜏 )
13	10 12	sylbir	⊢ ( 𝜒 → 𝜏 )