Theorem nprofelimr-P4.6b 391

Description: Negated Process of Elimination (right). †

Hypotheses

Ref	Expression
nprofelimr-P4.6b.1	⊢ (𝛾 → ¬ (𝜑 ∧ 𝜓))
nprofelimr-P4.6b.2	⊢ (𝛾 → 𝜓)

Assertion

Ref	Expression
nprofelimr-P4.6b	⊢ (𝛾 → ¬ 𝜑)

Proof of Theorem nprofelimr-P4.6b

Step	Hyp	Ref	Expression
1		rcp-NDASM2of2 194	. . 3 ⊢ ((𝛾 ∧ 𝜑) → 𝜑)
2		nprofelimr-P4.6b.2	. . . 4 ⊢ (𝛾 → 𝜓)
3	2	rcp-NDIMP1add1 208	. . 3 ⊢ ((𝛾 ∧ 𝜑) → 𝜓)
4	1, 3	ndandi-P3.7 172	. 2 ⊢ ((𝛾 ∧ 𝜑) → (𝜑 ∧ 𝜓))
5		nprofelimr-P4.6b.1	. . 3 ⊢ (𝛾 → ¬ (𝜑 ∧ 𝜓))
6	5	rcp-NDIMP1add1 208	. 2 ⊢ ((𝛾 ∧ 𝜑) → ¬ (𝜑 ∧ 𝜓))
7	4, 6	rcp-NDNEGI2 219	1 ⊢ (𝛾 → ¬ 𝜑)

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

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

This theorem is referenced by:  nprofelimr-P4.6b.RC  392