Theorem syl3an9b 1300

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 458	. . 3 ⊢ ((𝜑 ∧ 𝜃) → (𝜓 ↔ 𝜏))
4		syl3an9b.3	. . 3 ⊢ (𝜂 → (𝜏 ↔ 𝜁))
5	3, 4	sylan9bb 458	. 2 ⊢ (((𝜑 ∧ 𝜃) ∧ 𝜂) → (𝜓 ↔ 𝜁))
6	5	3impa 1184	1 ⊢ ((𝜑 ∧ 𝜃 ∧ 𝜂) → (𝜓 ↔ 𝜁))

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   ∧ w3a 968

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107

This theorem depends on definitions:  df-bi 116  df-3an 970

This theorem is referenced by:  eloprabg  5930