Theorem sepimandr-P4.9a 406

Description: Separate Right Conjunction from Implication. †

Hypothesis

Ref	Expression
sepimandr-P4.9a.1	⊢ (𝛾 → (𝜑 → (𝜓 ∧ 𝜒)))

Assertion

Ref	Expression
sepimandr-P4.9a	⊢ (𝛾 → ((𝜑 → 𝜓) ∧ (𝜑 → 𝜒)))

Proof of Theorem sepimandr-P4.9a

Step	Hyp	Ref	Expression
1		rcp-NDASM2of2 194	. . . . 5 ⊢ ((𝛾 ∧ 𝜑) → 𝜑)
2		sepimandr-P4.9a.1	. . . . . 6 ⊢ (𝛾 → (𝜑 → (𝜓 ∧ 𝜒)))
3	2	rcp-NDIMP1add1 208	. . . . 5 ⊢ ((𝛾 ∧ 𝜑) → (𝜑 → (𝜓 ∧ 𝜒)))
4	1, 3	ndime-P3.6 171	. . . 4 ⊢ ((𝛾 ∧ 𝜑) → (𝜓 ∧ 𝜒))
5	4	ndander-P3.9 174	. . 3 ⊢ ((𝛾 ∧ 𝜑) → 𝜓)
6	5	rcp-NDIMI2 224	. 2 ⊢ (𝛾 → (𝜑 → 𝜓))
7	4	ndandel-P3.8 173	. . 3 ⊢ ((𝛾 ∧ 𝜑) → 𝜒)
8	7	rcp-NDIMI2 224	. 2 ⊢ (𝛾 → (𝜑 → 𝜒))
9	6, 8	ndandi-P3.7 172	1 ⊢ (𝛾 → ((𝜑 → 𝜓) ∧ (𝜑 → 𝜒)))

Syntax hints:   → 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:  sepimandr-P4.9a.RC  407  sepimandr-P4.9a.CL  408