Theorem bibi1 343

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 335	1 ⊢ ((𝜑 ↔ 𝜓) → ((𝜑 ↔ 𝜒) ↔ (𝜓 ↔ 𝜒)))

Syntax hints:   → wi 4   ↔ wb 198

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 199

This theorem is referenced by:  bitr3  344  bitr  841  eqeq1d  2827  sbeqalb  3715  isclo2  21263  sbc3orgVD  39905  trsbcVD  39931  sbcssgVD  39937  csbingVD  39938  csbsngVD  39947  csbxpgVD  39948  csbrngVD  39950  csbunigVD  39952  csbfv12gALTVD  39953  e2ebindVD  39966