Theorem ndfalsee-P3.20 185

Description: Natural Deduction: '⊥' Elimination Rule.

Rules 19 and 20 allow us to write '(𝜑 → ⊥)' as a synonym for "'𝜑' is false" (or, more precisely, refutable).

Hypothesis

Ref	Expression
ndfalsee-P3.20.1	⊢ (𝜑 → ⊥)

Assertion

Ref	Expression
ndfalsee-P3.20	⊢ ¬ 𝜑

Proof of Theorem ndfalsee-P3.20

Step	Hyp	Ref	Expression
1		false-P2.15 159	. 2 ⊢ ¬ ⊥
2		ndfalsee-P3.20.1	. . 3 ⊢ (𝜑 → ⊥)
3	2	trnsp-P1.15c.SH 81	. 2 ⊢ (¬ ⊥ → ¬ 𝜑)
4	1, 3	ax-MP 14	1 ⊢ ¬ 𝜑

Syntax hints:  ¬ wff-neg 9   → wff-imp 10  ⊥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-true-D2.4 155  df-false-D2.5 158

This theorem is referenced by:  false-P3.45  353  falseimpoe-P4.4c  383  rcp-FALSENEGI1  433  truthtbltbif-P4.39b  508  truthtblfbit-P4.39c  509