Theorem bibi1 352

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 344	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  353  bitr  802  eqeq1d  2740  sbeqalb  3784  isclo2  22239  sbc3orgVD  42471  trsbcVD  42497  sbcssgVD  42503  csbingVD  42504  csbsngVD  42513  csbxpgVD  42514  csbrngVD  42516  csbunigVD  42518  csbfv12gALTVD  42519  e2ebindVD  42532