Theorem nmt-P3.32b 294

Description: Negated Modus Tollens.

This statement is the deductive form of clav-P1.12 68. It requires the Law of Excluded Middle and is thus not deducible with intuitionist logic.

Hypothesis

Ref	Expression
nmt-P3.32b.1	⊢ (𝛾 → (¬ 𝜑 → 𝜑))

Assertion

Ref	Expression
nmt-P3.32b	⊢ (𝛾 → 𝜑)

Proof of Theorem nmt-P3.32b

Step	Hyp	Ref	Expression
1		rcp-NDASM2of2 194	. . . 4 ⊢ ((𝛾 ∧ ¬ 𝜑) → ¬ 𝜑)
2		nmt-P3.32b.1	. . . . 5 ⊢ (𝛾 → (¬ 𝜑 → 𝜑))
3	2	rcp-NDIMP1add1 208	. . . 4 ⊢ ((𝛾 ∧ ¬ 𝜑) → (¬ 𝜑 → 𝜑))
4	1, 3	ndime-P3.6 171	. . 3 ⊢ ((𝛾 ∧ ¬ 𝜑) → 𝜑)
5	4, 1	rcp-NDNEGI2 219	. 2 ⊢ (𝛾 → ¬ ¬ 𝜑)
6	5	dnege-P3.30 276	1 ⊢ (𝛾 → 𝜑)

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  df-or-D2.3 145  df-true-D2.4 155

This theorem is referenced by:  nmt-P3.32b.RC  295  nmt-3.32b.CL  296