Theorem rcp-NDJOIN5 191

Description: ( ( 1 ∧ 2 ∧ 3 ∧ 4 ) ∧ 5 ) ⇒ ( 1 ∧ 2 ∧ 3 ∧ 4 ∧ 5 ).

Hypothesis

Ref	Expression
rcp-NDJOIN5.1	⊢ (((𝛾₁ ∧ 𝛾₂ ∧ 𝛾₃ ∧ 𝛾₄) ∧ 𝛾₅) → 𝜑)

Assertion

Ref	Expression
rcp-NDJOIN5	⊢ ((𝛾₁ ∧ 𝛾₂ ∧ 𝛾₃ ∧ 𝛾₄ ∧ 𝛾₅) → 𝜑)

Proof of Theorem rcp-NDJOIN5

Step	Hyp	Ref	Expression
1		rcp-NDJOIN5.1	. 2 ⊢ (((𝛾₁ ∧ 𝛾₂ ∧ 𝛾₃ ∧ 𝛾₄) ∧ 𝛾₅) → 𝜑)
2		df-rcp-AND5 165	. . . . 5 ⊢ ((𝛾₁ ∧ 𝛾₂ ∧ 𝛾₃ ∧ 𝛾₄ ∧ 𝛾₅) ↔ ((𝛾₁ ∧ 𝛾₂ ∧ 𝛾₃ ∧ 𝛾₄) ∧ 𝛾₅))
3	2	bisym-P2.6b.SH 125	. . . 4 ⊢ (((𝛾₁ ∧ 𝛾₂ ∧ 𝛾₃ ∧ 𝛾₄) ∧ 𝛾₅) ↔ (𝛾₁ ∧ 𝛾₂ ∧ 𝛾₃ ∧ 𝛾₄ ∧ 𝛾₅))
4	3	subiml-P2.8a.SH 129	. . 3 ⊢ ((((𝛾₁ ∧ 𝛾₂ ∧ 𝛾₃ ∧ 𝛾₄) ∧ 𝛾₅) → 𝜑) ↔ ((𝛾₁ ∧ 𝛾₂ ∧ 𝛾₃ ∧ 𝛾₄ ∧ 𝛾₅) → 𝜑))
5	4	bifwd-P2.5a.SH 112	. 2 ⊢ ((((𝛾₁ ∧ 𝛾₂ ∧ 𝛾₃ ∧ 𝛾₄) ∧ 𝛾₅) → 𝜑) → ((𝛾₁ ∧ 𝛾₂ ∧ 𝛾₃ ∧ 𝛾₄ ∧ 𝛾₅) → 𝜑))
6	1, 5	ax-MP 14	1 ⊢ ((𝛾₁ ∧ 𝛾₂ ∧ 𝛾₃ ∧ 𝛾₄ ∧ 𝛾₅) → 𝜑)

Syntax hints:   → wff-imp 10   ∧ wff-and 132   ∧ wff-rcp-AND4 162   ∧ wff-rcp-AND5 164

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-AND5 165

This theorem is referenced by:  rcp-NDASM1of5  202  rcp-NDASM2of5  203  rcp-NDASM3of5  204  rcp-NDASM4of5  205  rcp-NDASM5of5  206  rcp-NDIMP4add1  211  rcp-IMPIME4  530