Theorem bnj951 30100

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 1235	. 2 ⊢ (𝜏 → (𝜑 ∧ 𝜓 ∧ 𝜒))
5		bnj951.4	. 2 ⊢ (𝜏 → 𝜃)
6		df-bnj17 30006	. 2 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒 ∧ 𝜃) ↔ ((𝜑 ∧ 𝜓 ∧ 𝜒) ∧ 𝜃))
7	4, 5, 6	sylanbrc 695	1 ⊢ (𝜏 → (𝜑 ∧ 𝜓 ∧ 𝜒 ∧ 𝜃))

Syntax hints:   → wi 4   ∧ w3a 1031   ∧ w-bnj17 30005

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 196  df-an 385  df-3an 1033  df-bnj17 30006

This theorem is referenced by:  bnj966  30268  bnj967  30269  bnj910  30272  bnj1006  30283  bnj1118  30306  bnj1177  30328