Theorem rcp-NDJOIN3 189

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

Hypothesis

Ref	Expression
rcp-NDJOIN3.1	⊢ (((𝛾₁ ∧ 𝛾₂) ∧ 𝛾₃) → 𝜑)

Assertion

Ref	Expression
rcp-NDJOIN3	⊢ ((𝛾₁ ∧ 𝛾₂ ∧ 𝛾₃) → 𝜑)

Proof of Theorem rcp-NDJOIN3

Step	Hyp	Ref	Expression
1		rcp-NDJOIN3.1	. 2 ⊢ (((𝛾₁ ∧ 𝛾₂) ∧ 𝛾₃) → 𝜑)
2		df-rcp-AND3 161	. . . . 5 ⊢ ((𝛾₁ ∧ 𝛾₂ ∧ 𝛾₃) ↔ ((𝛾₁ ∧ 𝛾₂) ∧ 𝛾₃))
3	2	bisym-P2.6b.SH 125	. . . 4 ⊢ (((𝛾₁ ∧ 𝛾₂) ∧ 𝛾₃) ↔ (𝛾₁ ∧ 𝛾₂ ∧ 𝛾₃))
4	3	subiml-P2.8a.SH 129	. . 3 ⊢ ((((𝛾₁ ∧ 𝛾₂) ∧ 𝛾₃) → 𝜑) ↔ ((𝛾₁ ∧ 𝛾₂ ∧ 𝛾₃) → 𝜑))
5	4	bifwd-P2.5a.SH 112	. 2 ⊢ ((((𝛾₁ ∧ 𝛾₂) ∧ 𝛾₃) → 𝜑) → ((𝛾₁ ∧ 𝛾₂ ∧ 𝛾₃) → 𝜑))
6	1, 5	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-NDASM1of3  195  rcp-NDASM2of3  196  rcp-NDASM3of3  197  rcp-NDIMP2add1  209  rcp-IMPIME2  528