Theorem syl13anc 1184

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)
syl13anc.5	⊢ ((ψ ∧ (χ ∧ θ ∧ τ)) → η)

Assertion

Ref	Expression
syl13anc	⊢ (φ → η)

Proof of Theorem syl13anc

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		syl13anc.5	. 2 ⊢ ((ψ ∧ (χ ∧ θ ∧ τ)) → η)
7	1, 5, 6	syl2anc 642	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:  syl23anc  1189  syl33anc  1197  pm2.61da3ne  2597  sbthlem3  6206