Theorem biantr 858

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 19	. . 3 ⊢ ((χ ↔ ψ) → (χ ↔ ψ))
2	1	bibi2d 221	. 2 ⊢ ((χ ↔ ψ) → ((φ ↔ χ) ↔ (φ ↔ ψ)))
3	2	biimparc 283	1 ⊢ (((φ ↔ ψ) ∧ (χ ↔ ψ)) → (φ ↔ χ))

Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101

This theorem depends on definitions:  df-bi 110

This theorem is referenced by:  bm1.1  2022