Theorem bitrns-P2.6c 126

Description: Equivalence Property: '↔' Transitivity.

Assertion

Ref	Expression
bitrns-P2.6c	⊢ ((𝜑 ↔ 𝜓) → ((𝜓 ↔ 𝜒) → (𝜑 ↔ 𝜒)))

Proof of Theorem bitrns-P2.6c

Step	Hyp	Ref	Expression
1		ax-L1 11	. . . 4 ⊢ ((𝜑 ↔ 𝜓) → ((𝜓 ↔ 𝜒) → (𝜑 ↔ 𝜓)))
2	1	bifwd-P2.5a.2AC.SH 114	. . 3 ⊢ ((𝜑 ↔ 𝜓) → ((𝜓 ↔ 𝜒) → (𝜑 → 𝜓)))
3		id-P1.4 36	. . . . 5 ⊢ ((𝜓 ↔ 𝜒) → (𝜓 ↔ 𝜒))
4	3	axL1.SH 30	. . . 4 ⊢ ((𝜑 ↔ 𝜓) → ((𝜓 ↔ 𝜒) → (𝜓 ↔ 𝜒)))
5	4	bifwd-P2.5a.2AC.SH 114	. . 3 ⊢ ((𝜑 ↔ 𝜓) → ((𝜓 ↔ 𝜒) → (𝜓 → 𝜒)))
6	2, 5	sylt-P1.9.2AC.2SH 63	. 2 ⊢ ((𝜑 ↔ 𝜓) → ((𝜓 ↔ 𝜒) → (𝜑 → 𝜒)))
7	4	birev-P2.5b.2AC.SH 118	. . 3 ⊢ ((𝜑 ↔ 𝜓) → ((𝜓 ↔ 𝜒) → (𝜒 → 𝜓)))
8	1	birev-P2.5b.2AC.SH 118	. . 3 ⊢ ((𝜑 ↔ 𝜓) → ((𝜓 ↔ 𝜒) → (𝜓 → 𝜑)))
9	7, 8	sylt-P1.9.2AC.2SH 63	. 2 ⊢ ((𝜑 ↔ 𝜓) → ((𝜓 ↔ 𝜒) → (𝜒 → 𝜑)))
10	6, 9	bicmb-P2.5c.2AC.2SH 122	1 ⊢ ((𝜑 ↔ 𝜓) → ((𝜓 ↔ 𝜒) → (𝜑 ↔ 𝜒)))

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

This theorem is referenced by: (None)