Theorem mt-P3.32a 291

Description: Modus Tollens. †

This statement is the deductive form of nclav-P1.14 73. It does not rely on the Law of Excluded Middle, and is thus deducible with intuitionist logic.

Hypothesis

Ref	Expression
mt-P3.32a.1	⊢ (𝛾 → (𝜑 → ¬ 𝜑))

Assertion

Ref	Expression
mt-P3.32a	⊢ (𝛾 → ¬ 𝜑)

Proof of Theorem mt-P3.32a

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

This theorem is referenced by:  mt-P3.32a.RC  292  mt-3.32a.CL  293