Theorem axL2-P3.22 254

Description: Re-derived Deductive Form of Axiom L2. †

Hypothesis

Ref	Expression
axL2-P3.22.1	⊢ (𝛾 → (𝜑 → (𝜓 → 𝜒)))

Assertion

Ref	Expression
axL2-P3.22	⊢ (𝛾 → ((𝜑 → 𝜓) → (𝜑 → 𝜒)))

Proof of Theorem axL2-P3.22

Step	Hyp	Ref	Expression
1		rcp-NDASM3of3 197	. . . . 5 ⊢ ((𝛾 ∧ (𝜑 → 𝜓) ∧ 𝜑) → 𝜑)
2		rcp-NDASM2of3 196	. . . . 5 ⊢ ((𝛾 ∧ (𝜑 → 𝜓) ∧ 𝜑) → (𝜑 → 𝜓))
3	1, 2	ndime-P3.6 171	. . . 4 ⊢ ((𝛾 ∧ (𝜑 → 𝜓) ∧ 𝜑) → 𝜓)
4		axL2-P3.22.1	. . . . . 6 ⊢ (𝛾 → (𝜑 → (𝜓 → 𝜒)))
5	4	rcp-NDIMP1add2 212	. . . . 5 ⊢ ((𝛾 ∧ (𝜑 → 𝜓) ∧ 𝜑) → (𝜑 → (𝜓 → 𝜒)))
6	1, 5	ndime-P3.6 171	. . . 4 ⊢ ((𝛾 ∧ (𝜑 → 𝜓) ∧ 𝜑) → (𝜓 → 𝜒))
7	3, 6	ndime-P3.6 171	. . 3 ⊢ ((𝛾 ∧ (𝜑 → 𝜓) ∧ 𝜑) → 𝜒)
8	7	rcp-NDIMI3 225	. 2 ⊢ ((𝛾 ∧ (𝜑 → 𝜓)) → (𝜑 → 𝜒))
9	8	rcp-NDIMI2 224	1 ⊢ (𝛾 → ((𝜑 → 𝜓) → (𝜑 → 𝜒)))

Syntax hints:   → wff-imp 10   ∧ wff-rcp-AND3 160

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-rcp-AND3 161

This theorem is referenced by:  axL2-P3.22.RC  255  axL2-P3.22.CL  256  psubim-P6-L1  789