Theorem syl131anc 1195

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)
syl131anc.6	⊢ ((ψ ∧ (χ ∧ θ ∧ τ) ∧ η) → ζ)

Assertion

Ref	Expression
syl131anc	⊢ (φ → ζ)

Proof of Theorem syl131anc

Step	Hyp	Ref	Expression
1		sylXanc.1	. 2 ⊢ (φ → ψ)
2		sylXanc.2	. . 3 ⊢ (φ → χ)
3		sylXanc.3	. . 3 ⊢ (φ → θ)
4		sylXanc.4	. . 3 ⊢ (φ → τ)
5	2, 3, 4	3jca 1132	. 2 ⊢ (φ → (χ ∧ θ ∧ τ))
6		sylXanc.5	. 2 ⊢ (φ → η)
7		syl131anc.6	. 2 ⊢ ((ψ ∧ (χ ∧ θ ∧ τ) ∧ η) → ζ)
8	1, 5, 6, 7	syl3anc 1182	1 ⊢ (φ → ζ)

This proof depends on syntax axioms:   → wi 4   ∧ w3a 934

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This proof depends on definitions:  df-bi 177  df-an 360  df-3an 936

This theorem is used by:  syl132anc  1200  syl231anc  1202  syl133anc  1205