Theorem bicomdALT 40164

Description: Alternate proof of bicomd 222 which is shorter after expanding all parent theorems (as of 8-Aug-2024, bicom 221 depends on bicom1 220 and sylib 217 depends on syl 17). Additionally, the labels bicom1 220 and syl 17 happen to contain fewer characters than bicom 221 and sylib 217. However, neither of these conditions count as a shortening according to conventions 28764. 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 919 and pm2.31 920). 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 220	. 2 ⊢ ((𝜓 ↔ 𝜒) → (𝜒 ↔ 𝜓))
3	1, 2	syl 17	1 ⊢ (𝜑 → (𝜒 ↔ 𝜓))

Syntax hints:   → wi 4   ↔ wb 205

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 206

This theorem is referenced by: (None)