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 207

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 208

This theorem is referenced by:  bitr3  353  bitr  810  eqeq1d  2741  sbeqalb  3785  isclo2  23071  sbc3orgVD  45294  trsbcVD  45320  sbcssgVD  45326  csbingVD  45327  csbsngVD  45336  csbxpgVD  45337  csbrngVD  45339  csbunigVD  45341  csbfv12gALTVD  45342  e2ebindVD  45355