Theorem syl313anc 1206

Description: Syllogism combined with contraction. (Contributed by NM, 11-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)
syl313anc.8	⊢ (((ψ ∧ χ ∧ θ) ∧ τ ∧ (η ∧ ζ ∧ σ)) → ρ)

Assertion

Ref	Expression
syl313anc	⊢ (φ → ρ)

Proof of Theorem syl313anc

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	. . 3 ⊢ (φ → η)
6		sylXanc.6	. . 3 ⊢ (φ → ζ)
7		sylXanc.7	. . 3 ⊢ (φ → σ)
8	5, 6, 7	3jca 1132	. 2 ⊢ (φ → (η ∧ ζ ∧ σ))
9		syl313anc.8	. 2 ⊢ (((ψ ∧ χ ∧ θ) ∧ τ ∧ (η ∧ ζ ∧ σ)) → ρ)
10	1, 2, 3, 4, 8, 9	syl311anc 1196	1 ⊢ (φ → ρ)

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:  syl323anc  1212