Theorem mae-P3.23 257

Description: Middle Antecedent Elimination. †

Hypotheses

Ref	Expression
mae-P3.23.1	⊢ (𝛾 → (𝜑 → (𝜓 → 𝜒)))
mae-P3.23.2	⊢ (𝛾 → 𝜓)

Assertion

Ref	Expression
mae-P3.23	⊢ (𝛾 → (𝜑 → 𝜒))

Proof of Theorem mae-P3.23

Step	Hyp	Ref	Expression
1		mae-P3.23.2	. . . 4 ⊢ (𝛾 → 𝜓)
2	1	rcp-NDIMP1add1 208	. . 3 ⊢ ((𝛾 ∧ 𝜑) → 𝜓)
3		rcp-NDASM2of2 194	. . . 4 ⊢ ((𝛾 ∧ 𝜑) → 𝜑)
4		mae-P3.23.1	. . . . 5 ⊢ (𝛾 → (𝜑 → (𝜓 → 𝜒)))
5	4	rcp-NDIMP1add1 208	. . . 4 ⊢ ((𝛾 ∧ 𝜑) → (𝜑 → (𝜓 → 𝜒)))
6	3, 5	ndime-P3.6 171	. . 3 ⊢ ((𝛾 ∧ 𝜑) → (𝜓 → 𝜒))
7	2, 6	ndime-P3.6 171	. 2 ⊢ ((𝛾 ∧ 𝜑) → 𝜒)
8	7	rcp-NDIMI2 224	1 ⊢ (𝛾 → (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:  mae-P3.23.RC  258  eqsym-P5  627  exi-P6  718