Theorem syl3an9b 1433

Description: Nested syllogism inference conjoining 3 dissimilar antecedents. (Contributed by NM, 1-May-1995.)

Hypotheses

Ref	Expression
syl3an9b.1	⊢ (𝜑 → (𝜓 ↔ 𝜒))
syl3an9b.2	⊢ (𝜃 → (𝜒 ↔ 𝜏))
syl3an9b.3	⊢ (𝜂 → (𝜏 ↔ 𝜁))

Assertion

Ref	Expression
syl3an9b	⊢ ((𝜑 ∧ 𝜃 ∧ 𝜂) → (𝜓 ↔ 𝜁))

Proof of Theorem syl3an9b

Step	Hyp	Ref	Expression
1		syl3an9b.1	. . . 4 ⊢ (𝜑 → (𝜓 ↔ 𝜒))
2		syl3an9b.2	. . . 4 ⊢ (𝜃 → (𝜒 ↔ 𝜏))
3	1, 2	sylan9bb 510	. . 3 ⊢ ((𝜑 ∧ 𝜃) → (𝜓 ↔ 𝜏))
4		syl3an9b.3	. . 3 ⊢ (𝜂 → (𝜏 ↔ 𝜁))
5	3, 4	sylan9bb 510	. 2 ⊢ (((𝜑 ∧ 𝜃) ∧ 𝜂) → (𝜓 ↔ 𝜁))
6	5	3impa 1109	1 ⊢ ((𝜑 ∧ 𝜃 ∧ 𝜂) → (𝜓 ↔ 𝜁))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   ∧ w3a 1086

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem depends on definitions:  df-bi 206  df-an 397  df-3an 1088

This theorem is referenced by:  eloprabg  7384  dihjatcclem4  39435