Theorem dbitrns-P4.16 428

Description: Doubled Transitive Rule for Biconditional. †

Hypotheses

Ref	Expression
dbitrns-P4.16.1	⊢ (𝛾 → (𝜑 ↔ 𝜓))
dbitrns-P4.16.2	⊢ (𝛾 → (𝜓 ↔ 𝜒))
dbitrns-P4.16.3	⊢ (𝛾 → (𝜒 ↔ 𝜗))

Assertion

Ref	Expression
dbitrns-P4.16	⊢ (𝛾 → (𝜑 ↔ 𝜗))

Proof of Theorem dbitrns-P4.16

Step	Hyp	Ref	Expression
1		dbitrns-P4.16.1	. . 3 ⊢ (𝛾 → (𝜑 ↔ 𝜓))
2		dbitrns-P4.16.2	. . 3 ⊢ (𝛾 → (𝜓 ↔ 𝜒))
3	1, 2	bitrns-P3.33c 302	. 2 ⊢ (𝛾 → (𝜑 ↔ 𝜒))
4		dbitrns-P4.16.3	. 2 ⊢ (𝛾 → (𝜒 ↔ 𝜗))
5	3, 4	bitrns-P3.33c 302	1 ⊢ (𝛾 → (𝜑 ↔ 𝜗))

Syntax hints:   → wff-imp 10   ↔ wff-bi 104

This theorem was proved from axioms:  ax-L1 11  ax-L2 12  ax-L3 13  ax-MP 14

This theorem depends on definitions:  df-bi-D2.1 107  df-and-D2.2 133

This theorem is referenced by:  dbitrns-P4.16.RC  429  tbitrns-P4.17  430