Theorem syl12anc 1180

Description: Syllogism combined with contraction. (Contributed by Jeff Hankins, 1-Aug-2009.)

Hypotheses

Ref	Expression
sylXanc.1	⊢ (φ → ψ)
sylXanc.2	⊢ (φ → χ)
sylXanc.3	⊢ (φ → θ)
syl12anc.4	⊢ ((ψ ∧ (χ ∧ θ)) → τ)

Assertion

Ref	Expression
syl12anc	⊢ (φ → τ)

Proof of Theorem syl12anc

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	jca32 521	. 2 ⊢ (φ → (ψ ∧ (χ ∧ θ)))
5		syl12anc.4	. 2 ⊢ ((ψ ∧ (χ ∧ θ)) → τ)
6	4, 5	syl 15	1 ⊢ (φ → τ)

Syntax hints:   → wi 4   ∧ wa 358

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

This theorem is referenced by:  syl22anc  1183  raaan  3657  raaanv  3658  nndisjeq  4429  prepeano4  4451  ssfin  4470  ncfinraise  4481  ncfinlower  4483  nnpweq  4523  peano4  4557  f1oiso2  5500  frecsuc  6322