Theorem profelimr-P4.5b 387

Description: Process of Elimination (right). †

Hypotheses

Ref	Expression
profelimr-P4.5b.1	⊢ (𝛾 → (𝜑 ∨ 𝜓))
profelimr-P4.5b.2	⊢ (𝛾 → ¬ 𝜓)

Assertion

Ref	Expression
profelimr-P4.5b	⊢ (𝛾 → 𝜑)

Proof of Theorem profelimr-P4.5b

Step	Hyp	Ref	Expression
1		rcp-NDASM2of2 194	. 2 ⊢ ((𝛾 ∧ 𝜑) → 𝜑)
2		rcp-NDASM2of2 194	. . 3 ⊢ ((𝛾 ∧ 𝜓) → 𝜓)
3		profelimr-P4.5b.2	. . . 4 ⊢ (𝛾 → ¬ 𝜓)
4	3	rcp-NDIMP1add1 208	. . 3 ⊢ ((𝛾 ∧ 𝜓) → ¬ 𝜓)
5	2, 4	ndnege-P3.4 169	. 2 ⊢ ((𝛾 ∧ 𝜓) → 𝜑)
6		profelimr-P4.5b.1	. 2 ⊢ (𝛾 → (𝜑 ∨ 𝜓))
7	1, 5, 6	rcp-NDORE2 235	1 ⊢ (𝛾 → 𝜑)

Syntax hints:  ¬ wff-neg 9   → wff-imp 10   ∧ wff-and 132   ∨ wff-or 144

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-or-D2.3 145

This theorem is referenced by:  profelimr-P4.5b.RC  388  falseprofelimr-P4.7b  395