Theorem rcp-NDNEGI3 220

Description: ¬ Introduction Recipe. †

Hypotheses

Ref	Expression
rcp-NDNEGI3.1	⊢ ((𝛾₁ ∧ 𝛾₂ ∧ 𝛾₃) → 𝜑)
rcp-NDNEGI3.2	⊢ ((𝛾₁ ∧ 𝛾₂ ∧ 𝛾₃) → ¬ 𝜑)

Assertion

Ref	Expression
rcp-NDNEGI3	⊢ ((𝛾₁ ∧ 𝛾₂) → ¬ 𝛾₃)

Proof of Theorem rcp-NDNEGI3

Step	Hyp	Ref	Expression
1		rcp-NDNEGI3.1	. . 3 ⊢ ((𝛾₁ ∧ 𝛾₂ ∧ 𝛾₃) → 𝜑)
2	1	rcp-NDSEP3 186	. 2 ⊢ (((𝛾₁ ∧ 𝛾₂) ∧ 𝛾₃) → 𝜑)
3		rcp-NDNEGI3.2	. . 3 ⊢ ((𝛾₁ ∧ 𝛾₂ ∧ 𝛾₃) → ¬ 𝜑)
4	3	rcp-NDSEP3 186	. 2 ⊢ (((𝛾₁ ∧ 𝛾₂) ∧ 𝛾₃) → ¬ 𝜑)
5	2, 4	ndnegi-P3.3 168	1 ⊢ ((𝛾₁ ∧ 𝛾₂) → ¬ 𝛾₃)

Syntax hints:  ¬ wff-neg 9   → 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-and-D2.2 133  df-rcp-AND3 161

This theorem is referenced by:  ndnegi-P3.3.CL  242  trnsp-P3.31a  279  trnsp-P3.31b  282  trnsp-P3.31c  285  trnsp-P3.31d  288  rcp-RAA3  517