Theorem falsenegi-P4.18 432

Description: Derived Natural Deduction Rule Using '⊥'. †

Hypothesis

Ref	Expression
falsenegi-P4.18.1	⊢ ((𝛾 ∧ 𝜑) → ⊥)

Assertion

Ref	Expression
falsenegi-P4.18	⊢ (𝛾 → ¬ 𝜑)

Proof of Theorem falsenegi-P4.18

Step	Hyp	Ref	Expression
1		falsenegi-P4.18.1	. 2 ⊢ ((𝛾 ∧ 𝜑) → ⊥)
2	1	falseimpoe-P4.4c 383	. 2 ⊢ ((𝛾 ∧ 𝜑) → ¬ ⊥)
3	1, 2	rcp-NDNEGI2 219	1 ⊢ (𝛾 → ¬ 𝜑)

Syntax hints:  ¬ wff-neg 9   → wff-imp 10   ∧ wff-and 132  ⊥wff-false 157

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-true-D2.4 155  df-false-D2.5 158

This theorem is referenced by:  rcp-FALSENEGI2  434  rcp-FALSENEGI3  435  rcp-FALSENEGI4  436  rcp-FALSENEGI5  437  falseie-P4.22b  445  falseraa-P4  520  allnegex-P7-L1  956  exnegallint-P7  1047