Theorem bnj771 31689

Description: ∧-manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)

Hypotheses

Ref	Expression
bnj771.1	⊢ (𝜂 ↔ (𝜑 ∧ 𝜓 ∧ 𝜒 ∧ 𝜃))
bnj771.2	⊢ (𝜒 → 𝜏)

Assertion

Ref	Expression
bnj771	⊢ (𝜂 → 𝜏)

Proof of Theorem bnj771

Step	Hyp	Ref	Expression
1		bnj771.1	. 2 ⊢ (𝜂 ↔ (𝜑 ∧ 𝜓 ∧ 𝜒 ∧ 𝜃))
2		bnj771.2	. . 3 ⊢ (𝜒 → 𝜏)
3	2	bnj707 31680	. 2 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒 ∧ 𝜃) → 𝜏)
4	1, 3	sylbi 209	1 ⊢ (𝜂 → 𝜏)

Syntax hints:   → wi 4   ↔ wb 198   ∧ w-bnj17 31610

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem depends on definitions:  df-bi 199  df-an 388  df-3an 1070  df-bnj17 31611

This theorem is referenced by:  bnj1247  31734  bnj996  31880  bnj1097  31904  bnj1145  31916  bnj1259  31939  bnj1296  31944  bnj1450  31973