Theorem trnsp-P3.31b 282

Description: Transposition Variant B (negation elimination).

This statement is the deductive form of trnsp-P1.15b 78. It is equivalent to Axiom L3 and requires the Law of Excluded Middle. It is thus not deducible with intuitionist logic.

Hypothesis

Ref	Expression
trnsp-P3.31b.1	⊢ (𝛾 → (¬ 𝜑 → 𝜓))

Assertion

Ref	Expression
trnsp-P3.31b	⊢ (𝛾 → (¬ 𝜓 → 𝜑))

Proof of Theorem trnsp-P3.31b

Step	Hyp	Ref	Expression
1		rcp-NDASM3of3 197	. . . . 5 ⊢ ((𝛾 ∧ ¬ 𝜓 ∧ ¬ 𝜑) → ¬ 𝜑)
2		trnsp-P3.31b.1	. . . . . 6 ⊢ (𝛾 → (¬ 𝜑 → 𝜓))
3	2	rcp-NDIMP1add2 212	. . . . 5 ⊢ ((𝛾 ∧ ¬ 𝜓 ∧ ¬ 𝜑) → (¬ 𝜑 → 𝜓))
4	1, 3	ndime-P3.6 171	. . . 4 ⊢ ((𝛾 ∧ ¬ 𝜓 ∧ ¬ 𝜑) → 𝜓)
5		rcp-NDASM2of3 196	. . . 4 ⊢ ((𝛾 ∧ ¬ 𝜓 ∧ ¬ 𝜑) → ¬ 𝜓)
6	4, 5	rcp-NDNEGI3 220	. . 3 ⊢ ((𝛾 ∧ ¬ 𝜓) → ¬ ¬ 𝜑)
7	6	dnege-P3.30 276	. 2 ⊢ ((𝛾 ∧ ¬ 𝜓) → 𝜑)
8	7	rcp-NDIMI2 224	1 ⊢ (𝛾 → (¬ 𝜓 → 𝜑))

Syntax hints:  ¬ wff-neg 9   → wff-imp 10   ∧ wff-and 132   ∧ 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-or-D2.3 145  df-true-D2.4 155  df-rcp-AND3 161

This theorem is referenced by:  trnsp-P3.31b.RC  283  trnsp-P3.31b.CL  284  nfrexgencl-L6  813