Theorem eelT11

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

	Ref	Expression
Hypotheses	eelT11.1	⊢ ( ⊤ → 𝜑 )
	eelT11.2	⊢ ( 𝜓 → 𝜒 )
	eelT11.3	⊢ ( 𝜓 → 𝜃 )
	eelT11.4	⊢ ( ( 𝜑 ∧ 𝜒 ∧ 𝜃 ) → 𝜏 )
Assertion	eelT11	⊢ ( 𝜓 → 𝜏 )

Proof

Step	Hyp	Ref	Expression
1		eelT11.1	⊢ ( ⊤ → 𝜑 )
2		eelT11.2	⊢ ( 𝜓 → 𝜒 )
3		eelT11.3	⊢ ( 𝜓 → 𝜃 )
4		eelT11.4	⊢ ( ( 𝜑 ∧ 𝜒 ∧ 𝜃 ) → 𝜏 )
5		3anass	⊢ ( ( ⊤ ∧ 𝜓 ∧ 𝜓 ) ↔ ( ⊤ ∧ ( 𝜓 ∧ 𝜓 ) ) )
6		truan	⊢ ( ( ⊤ ∧ ( 𝜓 ∧ 𝜓 ) ) ↔ ( 𝜓 ∧ 𝜓 ) )
7		anidm	⊢ ( ( 𝜓 ∧ 𝜓 ) ↔ 𝜓 )
8	5 6 7	3bitri	⊢ ( ( ⊤ ∧ 𝜓 ∧ 𝜓 ) ↔ 𝜓 )
9	1 4	syl3an1	⊢ ( ( ⊤ ∧ 𝜒 ∧ 𝜃 ) → 𝜏 )
10	2 9	syl3an2	⊢ ( ( ⊤ ∧ 𝜓 ∧ 𝜃 ) → 𝜏 )
11	3 10	syl3an3	⊢ ( ( ⊤ ∧ 𝜓 ∧ 𝜓 ) → 𝜏 )
12	8 11	sylbir	⊢ ( 𝜓 → 𝜏 )