Theorem syl31anc 1137

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 1083	. 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 391	1 ⊢ (φ → η)

Syntax hints:   → wi 4   ∧ wa 97   ∧ w3a 884

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101

This theorem depends on definitions:  df-bi 110  df-3an 886

This theorem is referenced by:  syl32anc  1142  stoic4b  1319  enq0tr  6417  ltmul12a  7607  lt2msq1  7632  ledivp1  7650  lemul1ad  7686  lemul2ad  7687  lediv2ad  8419  difelfznle  8763  expubnd  8965