Theorem bibi1 355

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 347	1 ⊢ ((𝜑 ↔ 𝜓) → ((𝜑 ↔ 𝜒) ↔ (𝜓 ↔ 𝜒)))

Syntax hints:   → wi 4   ↔ wb 209

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 210

This theorem is referenced by:  bitr3  356  bitr  804  eqeq1d  2800  sbeqalb  3783  isclo2  21693  sbc3orgVD  41557  trsbcVD  41583  sbcssgVD  41589  csbingVD  41590  csbsngVD  41599  csbxpgVD  41600  csbrngVD  41602  csbunigVD  41604  csbfv12gALTVD  41605  e2ebindVD  41618