Theorem trnsp-P3.31c 285

Description: Transposition Variant C (negation introduction). †

This statement is the deductive form of trnsp-P1.15c 80. It does not require the Law of Excluded Middle, and is thus deducible with intuitionist logic.

Hypothesis

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

Assertion

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

Proof of Theorem trnsp-P3.31c

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

Syntax hints:  ¬ wff-neg 9   → 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:  trnsp-P3.31c.RC  286  trnsp-P3.31c.CL  287  subneg-P3.39  323  sepimandl-P4.9d  415