Theorem bicomdALT 43247

Description: Alternate proof of bicomd 225 which is shorter after expanding all parent theorems (as of 8-Aug-2024, bicom 224 depends on bicom1 223 and sylib 220 depends on syl 17). Additionally, the labels bicom1 223 and syl 17 happen to contain fewer characters than bicom 224 and sylib 220. However, neither of these conditions count as a shortening according to conventions 30602. 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 932 and pm2.31 933). 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 223	. 2 ⊢ ((𝜓 ↔ 𝜒) → (𝜒 ↔ 𝜓))
3	1, 2	syl 17	1 ⊢ (𝜑 → (𝜒 ↔ 𝜓))

Syntax hints:   → wi 4   ↔ wb 208

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

This theorem is referenced by: (None)