Theorem syl31anc 1185

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)
syl31anc.5	⊢ (((ψ ∧ χ ∧ θ) ∧ τ) → η)

Assertion

Ref	Expression
syl31anc	⊢ (φ → η)

Proof of Theorem syl31anc

Step	Hyp	Ref	Expression
1		sylXanc.1	. . 3 ⊢ (φ → ψ)
2		sylXanc.2	. . 3 ⊢ (φ → χ)
3		sylXanc.3	. . 3 ⊢ (φ → θ)
4	1, 2, 3	3jca 1132	. 2 ⊢ (φ → (ψ ∧ χ ∧ θ))
5		sylXanc.4	. 2 ⊢ (φ → τ)
6		syl31anc.5	. 2 ⊢ (((ψ ∧ χ ∧ θ) ∧ τ) → η)
7	4, 5, 6	syl2anc 642	1 ⊢ (φ → η)

This proof depends on syntax axioms:   → wi 4   ∧ wa 358   ∧ 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:  syl32anc  1190  ltfinirr  4458  vfin1cltv  4548  leconnnc  6219  tlecg  6231