Theorem nimpoe-P4.4b 380

Description: Variation of Principle of Explosion Using Implication (negated form). †

Hypothesis

Ref	Expression
nimpoe-P4.4b.1	⊢ (𝛾 → 𝜑)

Assertion

Ref	Expression
nimpoe-P4.4b	⊢ (𝛾 → (¬ 𝜑 → 𝜓))

Proof of Theorem nimpoe-P4.4b

Step	Hyp	Ref	Expression
1		nimpoe-P4.4b.1	. . . 4 ⊢ (𝛾 → 𝜑)
2	1	rcp-NDIMP1add1 208	. . 3 ⊢ ((𝛾 ∧ ¬ 𝜑) → 𝜑)
3		rcp-NDASM2of2 194	. . 3 ⊢ ((𝛾 ∧ ¬ 𝜑) → ¬ 𝜑)
4	2, 3	ndnege-P3.4 169	. 2 ⊢ ((𝛾 ∧ ¬ 𝜑) → 𝜓)
5	4	rcp-NDIMI2 224	1 ⊢ (𝛾 → (¬ 𝜑 → 𝜓))

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:  nimpoe-P4.4b.RC  381  nimpoe-P4.4b.CL  382  exnegallint-P7  1047