Theorem imcomm-P3.27 265

Description: Commutation of Antecedents. †

Hypothesis

Ref	Expression
imcomm-P3.27.1	⊢ (𝛾 → (𝜑 → (𝜓 → 𝜒)))

Assertion

Ref	Expression
imcomm-P3.27	⊢ (𝛾 → (𝜓 → (𝜑 → 𝜒)))

Proof of Theorem imcomm-P3.27

Step	Hyp	Ref	Expression
1		rcp-NDASM2of3 196	. . . 4 ⊢ ((𝛾 ∧ 𝜓 ∧ 𝜑) → 𝜓)
2		rcp-NDASM3of3 197	. . . . 5 ⊢ ((𝛾 ∧ 𝜓 ∧ 𝜑) → 𝜑)
3		imcomm-P3.27.1	. . . . . 6 ⊢ (𝛾 → (𝜑 → (𝜓 → 𝜒)))
4	3	rcp-NDIMP1add2 212	. . . . 5 ⊢ ((𝛾 ∧ 𝜓 ∧ 𝜑) → (𝜑 → (𝜓 → 𝜒)))
5	2, 4	ndime-P3.6 171	. . . 4 ⊢ ((𝛾 ∧ 𝜓 ∧ 𝜑) → (𝜓 → 𝜒))
6	1, 5	ndime-P3.6 171	. . 3 ⊢ ((𝛾 ∧ 𝜓 ∧ 𝜑) → 𝜒)
7	6	rcp-NDIMI3 225	. 2 ⊢ ((𝛾 ∧ 𝜓) → (𝜑 → 𝜒))
8	7	rcp-NDIMI2 224	1 ⊢ (𝛾 → (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:  imcomm-P3.27.RC  266  imoverim-P4.30-L1  476