Theorem biantr 802

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 342	. 2 ⊢ ((𝜒 ↔ 𝜓) → ((𝜑 ↔ 𝜒) ↔ (𝜑 ↔ 𝜓)))
3	2	biimparc 479	1 ⊢ (((𝜑 ↔ 𝜓) ∧ (𝜒 ↔ 𝜓)) → (𝜑 ↔ 𝜒))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395

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 206  df-an 396

This theorem is referenced by:  axextmo  2713  bitr3VD  42358  sbcoreleleqVD  42368  trsbcVD  42386  sbcssgVD  42392