Theorem biantr 556

Description: A transitive law of equivalence. Compare Theorem *4.22 of [WhiteheadRussell] p. 117.

Assertion

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

Proof of Theorem biantr

Step	Hyp	Ref	Expression
1		id 9	. . 3 ⊢ ((χ ↔ ψ) → (χ ↔ ψ))
2	1	bibi2d 470	. 2 ⊢ ((χ ↔ ψ) → ((φ ↔ χ) ↔ (φ ↔ ψ)))
3	2	biimparc 327	1 ⊢ (((φ ↔ ψ) ∧ (χ ↔ ψ)) → (φ ↔ χ))

Syntax hints:   → wi 2   ↔ wb 127   ∧ wa 196

This theorem is referenced by:  bm1.1 1088

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6

This theorem depends on definitions:  df-bi 128  df-an 198

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