Theorem biantr 897

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 309	. 2 ⊢ ((χ ↔ ψ) → ((φ ↔ χ) ↔ (φ ↔ ψ)))
3	2	biimparc 473	1 ⊢ (((φ ↔ ψ) ∧ (χ ↔ ψ)) → (φ ↔ χ))

This proof depends on syntax axioms:   → wi 4   ↔ wb 176   ∧ wa 358

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This proof depends on definitions:  df-bi 177  df-an 360

This theorem is used by:  bm1.1  2338