Theorem negbicancel-P4.11 419

Description: Negation Cancellation Rule.

Hypothesis

Ref	Expression
negbicancel-P4.11.1	⊢ (𝛾 → (¬ 𝜑 ↔ ¬ 𝜓))

Assertion

Ref	Expression
negbicancel-P4.11	⊢ (𝛾 → (𝜑 ↔ 𝜓))

Proof of Theorem negbicancel-P4.11

Step	Hyp	Ref	Expression
1		negbicancel-P4.11.1	. . 3 ⊢ (𝛾 → (¬ 𝜑 ↔ ¬ 𝜓))
2	1	subneg-P3.39 323	. 2 ⊢ (𝛾 → (¬ ¬ 𝜑 ↔ ¬ ¬ 𝜓))
3		dnegeq-P4.10 418	. . . . 5 ⊢ (¬ ¬ 𝜑 ↔ 𝜑)
4	3	rcp-NDIMP0addall 207	. . . 4 ⊢ (𝛾 → (¬ ¬ 𝜑 ↔ 𝜑))
5		dnegeq-P4.10 418	. . . . 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-or-D2.3 145  df-true-D2.4 155  df-rcp-AND3 161

This theorem is referenced by:  negbicancel-P4.11.RC  420