Theorem syl3an9b 1250

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 680	. . 3 ⊢ ((φ ∧ θ) → (ψ ↔ τ))
4		syl3an9b.3	. . 3 ⊢ (η → (τ ↔ ζ))
5	3, 4	sylan9bb 680	. 2 ⊢ (((φ ∧ θ) ∧ η) → (ψ ↔ ζ))
6	5	3impa 1146	1 ⊢ ((φ ∧ θ ∧ η) → (ψ ↔ ζ))

Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358   ∧ w3a 934

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 177  df-an 360  df-3an 936

This theorem is referenced by:  eloprabg  5580