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Metamath Proof Explorer

Theorem 3bitr4d

Description: Deduction from transitivity of biconditional. Useful for converting conditional definitions in a formula. (Contributed by NM, 18-Oct-1995)

	Ref	Expression
Hypotheses	3bitr4d.1	⊢ ( 𝜑 → ( 𝜓 ↔ 𝜒 ) )
	3bitr4d.2	⊢ ( 𝜑 → ( 𝜃 ↔ 𝜓 ) )
	3bitr4d.3	⊢ ( 𝜑 → ( 𝜏 ↔ 𝜒 ) )
Assertion	3bitr4d	⊢ ( 𝜑 → ( 𝜃 ↔ 𝜏 ) )

Proof

Step	Hyp	Ref	Expression
1		3bitr4d.1	⊢ ( 𝜑 → ( 𝜓 ↔ 𝜒 ) )
2		3bitr4d.2	⊢ ( 𝜑 → ( 𝜃 ↔ 𝜓 ) )
3		3bitr4d.3	⊢ ( 𝜑 → ( 𝜏 ↔ 𝜒 ) )
4	1 3	bitr4d	⊢ ( 𝜑 → ( 𝜓 ↔ 𝜏 ) )
5	2 4	bitrd	⊢ ( 𝜑 → ( 𝜃 ↔ 𝜏 ) )