Theorem syl211anc 1188

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)
syl211anc.5	⊢ (((ψ ∧ χ) ∧ θ ∧ τ) → η)

Assertion

Ref	Expression
syl211anc	⊢ (φ → η)

Proof of Theorem syl211anc

Step	Hyp	Ref	Expression
1		sylXanc.1	. . 3 ⊢ (φ → ψ)
2		sylXanc.2	. . 3 ⊢ (φ → χ)
3	1, 2	jca 518	. 2 ⊢ (φ → (ψ ∧ χ))
4		sylXanc.3	. 2 ⊢ (φ → θ)
5		sylXanc.4	. 2 ⊢ (φ → τ)
6		syl211anc.5	. 2 ⊢ (((ψ ∧ χ) ∧ θ ∧ τ) → η)
7	3, 4, 5, 6	syl3anc 1182	1 ⊢ (φ → η)

Syntax hints:   → wi 4   ∧ 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:  syl212anc  1192  syl221anc  1193