Theorem aibnbaif 47704

Description: Given a implies b, not b, there exists a proof for a is F. (Contributed by Jarvin Udandy, 1-Sep-2016.)

Hypotheses

Ref	Expression
aibnbaif.1	⊢ (𝜑 → 𝜓)
aibnbaif.2	⊢ ¬ 𝜓

Assertion

Ref	Expression
aibnbaif	⊢ (𝜑 ↔ ⊥)

Proof of Theorem aibnbaif

Step	Hyp	Ref	Expression
1		aibnbaif.1	. . 3 ⊢ (𝜑 → 𝜓)
2		aibnbaif.2	. . 3 ⊢ ¬ 𝜓
3	1, 2	aibnbna 47703	. 2 ⊢ ¬ 𝜑
4	3	bifal 1586	1 ⊢ (𝜑 ↔ ⊥)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209  ⊥wfal 1582

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This proof depends on definitions:  df-bi 210  df-tru 1573  df-fal 1583

This theorem is used by:  conimpf  47714  conimpfalt  47715  dandysum2p2e4  47795