Theorem negbicancelint-P4.14 424

Description: Negation Cancellation Rule derivalbe with intuitionist logic. †

Hypothesis

Ref	Expression
negbicancelint-P4.14.1	⊢ (𝛾 → (¬ ¬ 𝜑 ↔ ¬ ¬ 𝜓))

Assertion

Ref	Expression
negbicancelint-P4.14	⊢ (𝛾 → (¬ 𝜑 ↔ ¬ 𝜓))

Proof of Theorem negbicancelint-P4.14

Step	Hyp	Ref	Expression
1		negbicancelint-P4.14.1	. . 3 ⊢ (𝛾 → (¬ ¬ 𝜑 ↔ ¬ ¬ 𝜓))
2	1	subneg-P3.39 323	. 2 ⊢ (𝛾 → (¬ ¬ ¬ 𝜑 ↔ ¬ ¬ ¬ 𝜓))
3		dnegeqint-P4.13 423	. . . . 5 ⊢ (¬ ¬ ¬ 𝜑 ↔ ¬ 𝜑)
4	3	rcp-NDIMP0addall 207	. . . 4 ⊢ (𝛾 → (¬ ¬ ¬ 𝜑 ↔ ¬ 𝜑))
5		dnegeqint-P4.13 423	. . . . 5 ⊢ (¬ ¬ ¬ 𝜓 ↔ ¬ 𝜓)
6	5	rcp-NDIMP0addall 207	. . . 4 ⊢ (𝛾 → (¬ ¬ ¬ 𝜓 ↔ ¬ 𝜓))
7	4, 6	subbid-P3.41c 336	. . 3 ⊢ (𝛾 → ((¬ ¬ ¬ 𝜑 ↔ ¬ ¬ ¬ 𝜓) ↔ (¬ 𝜑 ↔ ¬ 𝜓)))
8	7	ndbief-P3.14 179	. 2 ⊢ (𝛾 → ((¬ ¬ ¬ 𝜑 ↔ ¬ ¬ ¬ 𝜓) → (¬ 𝜑 ↔ ¬ 𝜓)))
9	2, 8	ndime-P3.6 171	1 ⊢ (𝛾 → (¬ 𝜑 ↔ ¬ 𝜓))

Syntax hints:  ¬ wff-neg 9   → wff-imp 10   ↔ wff-bi 104

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-true-D2.4 155  df-rcp-AND3 161

This theorem is referenced by:  negbicancelint-P4.14.RC  425