Theorem bibi1 350

Description: Theorem *4.86 of [WhiteheadRussell] p. 122. (Contributed by NM, 3-Jan-2005.)

Assertion

Ref	Expression
bibi1	⊢ ((𝜑 ↔ 𝜓) → ((𝜑 ↔ 𝜒) ↔ (𝜓 ↔ 𝜒)))

Proof of Theorem bibi1

Step	Hyp	Ref	Expression
1		id 22	. 2 ⊢ ((𝜑 ↔ 𝜓) → (𝜑 ↔ 𝜓))
2	1	bibi1d 342	1 ⊢ ((𝜑 ↔ 𝜓) → ((𝜑 ↔ 𝜒) ↔ (𝜓 ↔ 𝜒)))

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:  bitr3  351  bitr  801  eqeq1d  2732  sbeqalb  3846  isclo2  22814  sbc3orgVD  43916  trsbcVD  43942  sbcssgVD  43948  csbingVD  43949  csbsngVD  43958  csbxpgVD  43959  csbrngVD  43961  csbunigVD  43963  csbfv12gALTVD  43964  e2ebindVD  43977