Theorem syl222anc 1198

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)
syl222anc.7	⊢ (((ψ ∧ χ) ∧ (θ ∧ τ) ∧ (η ∧ ζ)) → σ)

Assertion

Ref	Expression
syl222anc	⊢ (φ → σ)

Proof of Theorem syl222anc

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	5, 6	jca 518	. 2 ⊢ (φ → (η ∧ ζ))
8		syl222anc.7	. 2 ⊢ (((ψ ∧ χ) ∧ (θ ∧ τ) ∧ (η ∧ ζ)) → σ)
9	1, 2, 3, 4, 7, 8	syl221anc 1193	1 ⊢ (φ → σ)

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:  3anandis  1283  3anandirs  1284