Theorem had1OLD 1635

Description: Obsolete version of had1 1633 as of 10-Aug-2026. (Contributed by Mario Carneiro, 4-Sep-2016.) (Proof shortened by Wolf Lammen, 11-Jul-2020.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
had1OLD	⊢ (𝜑 → (hadd(𝜑, 𝜓, 𝜒) ↔ (𝜓 ↔ 𝜒)))

Proof of Theorem had1OLD

Step	Hyp	Ref	Expression
1		hadrot 1631	. . . 4 ⊢ (hadd(𝜑, 𝜓, 𝜒) ↔ hadd(𝜓, 𝜒, 𝜑))
2		hadbi 1628	. . . 4 ⊢ (hadd(𝜓, 𝜒, 𝜑) ↔ ((𝜓 ↔ 𝜒) ↔ 𝜑))
3	1, 2	bitri 278	. . 3 ⊢ (hadd(𝜑, 𝜓, 𝜒) ↔ ((𝜓 ↔ 𝜒) ↔ 𝜑))
4		biass 388	. . 3 ⊢ (((hadd(𝜑, 𝜓, 𝜒) ↔ (𝜓 ↔ 𝜒)) ↔ 𝜑) ↔ (hadd(𝜑, 𝜓, 𝜒) ↔ ((𝜓 ↔ 𝜒) ↔ 𝜑)))
5	3, 4	mpbir 234	. 2 ⊢ ((hadd(𝜑, 𝜓, 𝜒) ↔ (𝜓 ↔ 𝜒)) ↔ 𝜑)
6	5	biimpri 231	1 ⊢ (𝜑 → (hadd(𝜑, 𝜓, 𝜒) ↔ (𝜓 ↔ 𝜒)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209  haddwhad 1623

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-xor 1542  df-had 1624

This theorem is used by:  had0OLD  1636  hadifpOLD  1638