Theorem bibi1 341

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 333	1 ⊢ ((𝜑 ↔ 𝜓) → ((𝜑 ↔ 𝜒) ↔ (𝜓 ↔ 𝜒)))

Syntax hints:   → wi 4   ↔ wb 196

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 197

This theorem is referenced by:  bitr3  342  bitr  745  eqeq1d  2624  sbeqalb  3488  isclo2  20892  sbc3orgVD  39086  trsbcVD  39113  sbcssgVD  39119  csbingVD  39120  csbsngVD  39129  csbxpgVD  39130  csbrngVD  39132  csbunigVD  39134  csbfv12gALTVD  39135  e2ebindVD  39148