Theorem aifftbifffaibif 43164

Description: Given a is equivalent to T., Given b is equivalent to F., there exists a proof for that a implies b is false. (Contributed by Jarvin Udandy, 7-Sep-2020.)

Hypotheses

Ref	Expression
aifftbifffaibif.1	⊢ (𝜑 ↔ ⊤)
aifftbifffaibif.2	⊢ (𝜓 ↔ ⊥)

Assertion

Ref	Expression
aifftbifffaibif	⊢ ((𝜑 → 𝜓) ↔ ⊥)

Proof of Theorem aifftbifffaibif

Step	Hyp	Ref	Expression
1		aifftbifffaibif.1	. . . . 5 ⊢ (𝜑 ↔ ⊤)
2	1	aistia 43140	. . . 4 ⊢ 𝜑
3		aifftbifffaibif.2	. . . . 5 ⊢ (𝜓 ↔ ⊥)
4	3	aisfina 43141	. . . 4 ⊢ ¬ 𝜓
5	2, 4	pm3.2i 473	. . 3 ⊢ (𝜑 ∧ ¬ 𝜓)
6		annim 406	. . . 4 ⊢ ((𝜑 ∧ ¬ 𝜓) ↔ ¬ (𝜑 → 𝜓))
7	6	biimpi 218	. . 3 ⊢ ((𝜑 ∧ ¬ 𝜓) → ¬ (𝜑 → 𝜓))
8	5, 7	ax-mp 5	. 2 ⊢ ¬ (𝜑 → 𝜓)
9	8	bifal 1553	1 ⊢ ((𝜑 → 𝜓) ↔ ⊥)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∧ wa 398  ⊤wtru 1538  ⊥wfal 1549

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem depends on definitions:  df-bi 209  df-an 399  df-tru 1540  df-fal 1550

This theorem is referenced by:  atnaiana  43166