Theorem rcp-NDSEP3 186

Description: ( 1 ∧ 2 ∧ 3 ) ⇒ ( ( 1 ∧ 2 ) ∧ 3 ).

Hypothesis

Ref	Expression
rcp-NDSEP3.1	⊢ ((𝛾₁ ∧ 𝛾₂ ∧ 𝛾₃) → 𝜑)

Assertion

Ref	Expression
rcp-NDSEP3	⊢ (((𝛾₁ ∧ 𝛾₂) ∧ 𝛾₃) → 𝜑)

Proof of Theorem rcp-NDSEP3

Step	Hyp	Ref	Expression
1		rcp-NDSEP3.1	. 2 ⊢ ((𝛾₁ ∧ 𝛾₂ ∧ 𝛾₃) → 𝜑)
2		df-rcp-AND3 161	. . . 4 ⊢ ((𝛾₁ ∧ 𝛾₂ ∧ 𝛾₃) ↔ ((𝛾₁ ∧ 𝛾₂) ∧ 𝛾₃))
3	2	subiml-P2.8a.SH 129	. . 3 ⊢ (((𝛾₁ ∧ 𝛾₂ ∧ 𝛾₃) → 𝜑) ↔ (((𝛾₁ ∧ 𝛾₂) ∧ 𝛾₃) → 𝜑))
4	3	bifwd-P2.5a.SH 112	. 2 ⊢ (((𝛾₁ ∧ 𝛾₂ ∧ 𝛾₃) → 𝜑) → (((𝛾₁ ∧ 𝛾₂) ∧ 𝛾₃) → 𝜑))
5	1, 4	ax-MP 14	1 ⊢ (((𝛾₁ ∧ 𝛾₂) ∧ 𝛾₃) → 𝜑)

Syntax hints:   → wff-imp 10   ∧ wff-and 132   ∧ wff-rcp-AND3 160

This theorem was proved from axioms:  ax-L1 11  ax-L2 12  ax-L3 13  ax-MP 14

This theorem depends on definitions:  df-bi-D2.1 107  df-rcp-AND3 161

This theorem is referenced by:  rcp-NDNEGI3  220  rcp-NDIMI3  225  rcp-NDORE3  236  rcp-FALSENEGI3  435