Theorem biantr 972

Description: A transitive law of equivalence. Compare Theorem *4.22 of [WhiteheadRussell] p. 117. (Contributed by NM, 18-Aug-1993.)

Assertion

Ref	Expression
biantr	⊢ (((𝜑 ↔ 𝜓) ∧ (𝜒 ↔ 𝜓)) → (𝜑 ↔ 𝜒))

Proof of Theorem biantr

Step	Hyp	Ref	Expression
1		id 22	. . 3 ⊢ ((𝜒 ↔ 𝜓) → (𝜒 ↔ 𝜓))
2	1	bibi2d 332	. 2 ⊢ ((𝜒 ↔ 𝜓) → ((𝜑 ↔ 𝜒) ↔ (𝜑 ↔ 𝜓)))
3	2	biimparc 504	1 ⊢ (((𝜑 ↔ 𝜓) ∧ (𝜒 ↔ 𝜓)) → (𝜑 ↔ 𝜒))

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384

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  df-an 386

This theorem is referenced by:  bm1.1  2607  bitr3VD  39084  sbcoreleleqVD  39095  trsbcVD  39113  sbcssgVD  39119