Theorem syl333anc 1214

Description: A syllogism inference combined with contraction. (Contributed by NM, 10-Mar-2012.)

Hypotheses

Ref	Expression
sylXanc.1	⊢ (φ → ψ)
sylXanc.2	⊢ (φ → χ)
sylXanc.3	⊢ (φ → θ)
sylXanc.4	⊢ (φ → τ)
sylXanc.5	⊢ (φ → η)
sylXanc.6	⊢ (φ → ζ)
sylXanc.7	⊢ (φ → σ)
sylXanc.8	⊢ (φ → ρ)
sylXanc.9	⊢ (φ → μ)
syl333anc.10	⊢ (((ψ ∧ χ ∧ θ) ∧ (τ ∧ η ∧ ζ) ∧ (σ ∧ ρ ∧ μ)) → λ)

Assertion

Ref	Expression
syl333anc	⊢ (φ → λ)

Proof of Theorem syl333anc

Step	Hyp	Ref	Expression
1		sylXanc.1	. 2 ⊢ (φ → ψ)
2		sylXanc.2	. 2 ⊢ (φ → χ)
3		sylXanc.3	. 2 ⊢ (φ → θ)
4		sylXanc.4	. 2 ⊢ (φ → τ)
5		sylXanc.5	. 2 ⊢ (φ → η)
6		sylXanc.6	. 2 ⊢ (φ → ζ)
7		sylXanc.7	. . 3 ⊢ (φ → σ)
8		sylXanc.8	. . 3 ⊢ (φ → ρ)
9		sylXanc.9	. . 3 ⊢ (φ → μ)
10	7, 8, 9	3jca 1132	. 2 ⊢ (φ → (σ ∧ ρ ∧ μ))
11		syl333anc.10	. 2 ⊢ (((ψ ∧ χ ∧ θ) ∧ (τ ∧ η ∧ ζ) ∧ (σ ∧ ρ ∧ μ)) → λ)
12	1, 2, 3, 4, 5, 6, 10, 11	syl331anc 1207	1 ⊢ (φ → λ)

Syntax hints:   → wi 4   ∧ 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: (None)