Theorem tbitrns-P4.17 430

Description: Triple Transitive Rule for Biconditional. †

Hypotheses

Ref	Expression
tbitrns-P4.17.1	⊢ (𝛾 → (𝜑 ↔ 𝜓))
tbitrns-P4.17.2	⊢ (𝛾 → (𝜓 ↔ 𝜒))
tbitrns-P4.17.3	⊢ (𝛾 → (𝜒 ↔ 𝜗))
tbitrns-P4.17.4	⊢ (𝛾 → (𝜗 ↔ 𝜏))

Assertion

Ref	Expression
tbitrns-P4.17	⊢ (𝛾 → (𝜑 ↔ 𝜏))

Proof of Theorem tbitrns-P4.17

Step	Hyp	Ref	Expression
1		tbitrns-P4.17.1	. . 3 ⊢ (𝛾 → (𝜑 ↔ 𝜓))
2		tbitrns-P4.17.2	. . 3 ⊢ (𝛾 → (𝜓 ↔ 𝜒))
3		tbitrns-P4.17.3	. . 3 ⊢ (𝛾 → (𝜒 ↔ 𝜗))
4	1, 2, 3	dbitrns-P4.16 428	. 2 ⊢ (𝛾 → (𝜑 ↔ 𝜗))
5		tbitrns-P4.17.4	. 2 ⊢ (𝛾 → (𝜗 ↔ 𝜏))
6	4, 5	bitrns-P3.33c 302	1 ⊢ (𝛾 → (𝜑 ↔ 𝜏))

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:  tbitrns-P4.17.RC  431