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)

This proof depends on syntax axioms:   → wi 4   ∧ wa 358

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

This theorem is used by:  syl22anc  1183  raaan  3658  raaanv  3659  nndisjeq  4430  prepeano4  4452  ssfin  4471  ncfinraise  4482  ncfinlower  4484  nnpweq  4524  peano4  4558  f1oiso2  5501  frecsuc  6323