Theorem nthmeqfalse-P4.21b 443

Description: Any Refuted Statement is Logically Equivalent to '⊥'. †

Hypothesis

Ref	Expression
nthmeqfalse-P4.21b.1	⊢ ¬ 𝜑

Assertion

Ref	Expression
nthmeqfalse-P4.21b	⊢ (𝜑 ↔ ⊥)

Proof of Theorem nthmeqfalse-P4.21b

Step	Hyp	Ref	Expression
1		nthmeqfalse-P4.21b.1	. . 3 ⊢ ¬ 𝜑
2	1	ndfalsei-P3.19 184	. 2 ⊢ (𝜑 → ⊥)
3		rcp-NDASM1of1 192	. . 3 ⊢ (⊥ → ⊥)
4	3	falseimpoe-P4.4c 383	. 2 ⊢ (⊥ → 𝜑)
5	2, 4	rcp-NDBII0 239	1 ⊢ (𝜑 ↔ ⊥)

Syntax hints:  ¬ wff-neg 9   ↔ wff-bi 104  ⊥wff-false 157

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-false-D2.5 158

This theorem is referenced by:  idornthml-P4.24a  448  idornthmr-P4.24b  449  truthtbltbif-P4.39b  508  truthtblfbit-P4.39c  509