Theorem bnj951 35406

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

Hypotheses

Ref	Expression
bnj951.1	⊢ (𝜏 → 𝜑)
bnj951.2	⊢ (𝜏 → 𝜓)
bnj951.3	⊢ (𝜏 → 𝜒)
bnj951.4	⊢ (𝜏 → 𝜃)

Assertion

Ref	Expression
bnj951	⊢ (𝜏 → (𝜑 ∧ 𝜓 ∧ 𝜒 ∧ 𝜃))

Proof of Theorem bnj951

Step	Hyp	Ref	Expression
1		bnj951.1	. . 3 ⊢ (𝜏 → 𝜑)
2		bnj951.2	. . 3 ⊢ (𝜏 → 𝜓)
3		bnj951.3	. . 3 ⊢ (𝜏 → 𝜒)
4	1, 2, 3	3jca 1146	. 2 ⊢ (𝜏 → (𝜑 ∧ 𝜓 ∧ 𝜒))
5		bnj951.4	. 2 ⊢ (𝜏 → 𝜃)
6		df-bnj17 35318	. 2 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒 ∧ 𝜃) ↔ ((𝜑 ∧ 𝜓 ∧ 𝜒) ∧ 𝜃))
7	4, 5, 6	sylanbrc 595	1 ⊢ (𝜏 → (𝜑 ∧ 𝜓 ∧ 𝜒 ∧ 𝜃))

This proof depends on syntax axioms:   → wi 4   ∧ w3a 1103   ∧ w-bnj17 35317

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This proof depends on definitions:  df-bi 210  df-an 402  df-3an 1105  df-bnj17 35318

This theorem is used by:  bnj966  35574  bnj967  35575  bnj910  35578  bnj1006  35590  bnj1118  35614  bnj1177  35636