Theorem aifftbifffaibifff 43165

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

Hypotheses

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

Assertion

Ref	Expression
aifftbifffaibifff	⊢ ((𝜑 ↔ 𝜓) ↔ ⊥)

Proof of Theorem aifftbifffaibifff

Step	Hyp	Ref	Expression
1		aifftbifffaibifff.1	. . . . 5 ⊢ (𝜑 ↔ ⊤)
2	1	aistia 43140	. . . 4 ⊢ 𝜑
3		aifftbifffaibifff.2	. . . . 5 ⊢ (𝜓 ↔ ⊥)
4	3	aisfina 43141	. . . 4 ⊢ ¬ 𝜓
5	2, 4	abnotbtaxb 43158	. . 3 ⊢ (𝜑 ⊻ 𝜓)
6	5	axorbtnotaiffb 43146	. 2 ⊢ ¬ (𝜑 ↔ 𝜓)
7		nbfal 1552	. . 3 ⊢ (¬ (𝜑 ↔ 𝜓) ↔ ((𝜑 ↔ 𝜓) ↔ ⊥))
8	7	biimpi 218	. 2 ⊢ (¬ (𝜑 ↔ 𝜓) → ((𝜑 ↔ 𝜓) ↔ ⊥))
9	6, 8	ax-mp 5	1 ⊢ ((𝜑 ↔ 𝜓) ↔ ⊥)

Syntax hints:  ¬ wn 3   ↔ wb 208  ⊤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-xor 1502  df-tru 1540  df-fal 1550

This theorem is referenced by: (None)