Theorem bicomdALT 42613

Description: Alternate proof of bicomd 223 which is shorter after expanding all parent theorems (as of 8-Aug-2024, bicom 222 depends on bicom1 221 and sylib 218 depends on syl 17). Additionally, the labels bicom1 221 and syl 17 happen to contain fewer characters than bicom 222 and sylib 218. However, neither of these conditions count as a shortening according to conventions 30424. In the first case, the criteria could easily be broken by upstream changes, and in many cases the upstream dependency tree is nontrivial (see orass 920 and pm2.31 921). For the latter case, theorem labels are up to revision, so they are not counted in the size of a proof. (Contributed by SN, 21-May-2022.) (New usage is discouraged.) (Proof modification is discouraged.)

Hypothesis

Ref	Expression
bicomdALT.1	⊢ (𝜑 → (𝜓 ↔ 𝜒))

Assertion

Ref	Expression
bicomdALT	⊢ (𝜑 → (𝜒 ↔ 𝜓))

Proof of Theorem bicomdALT

Step	Hyp	Ref	Expression
1		bicomdALT.1	. 2 ⊢ (𝜑 → (𝜓 ↔ 𝜒))
2		bicom1 221	. 2 ⊢ ((𝜓 ↔ 𝜒) → (𝜒 ↔ 𝜓))
3	1, 2	syl 17	1 ⊢ (𝜑 → (𝜒 ↔ 𝜓))

Syntax hints:   → wi 4   ↔ wb 206

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 207

This theorem is referenced by: (None)