Theorem profeliml-P4.5a 385

Description: Process of Elimination (left). †

Hypotheses

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

Assertion

Ref	Expression
profeliml-P4.5a	⊢ (𝛾 → 𝜓)

Proof of Theorem profeliml-P4.5a

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

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:  profeliml-P4.5a.RC  386  falseprofeliml-P4.7a  393  sepimorr-P4.9c  412  oroverim-P4.28-L2  466