Theorem sylsyld 52

Description: Virtual deduction rule e12 without virtual deduction symbols. (Contributed by Alan Sare, 20-Apr-2011.)

Hypotheses

Ref	Expression
sylsyld.1	⊢ (φ → ψ)
sylsyld.2	⊢ (φ → (χ → θ))
sylsyld.3	⊢ (ψ → (θ → τ))

Assertion

Ref	Expression
sylsyld	⊢ (φ → (χ → τ))

Proof of Theorem sylsyld

Step	Hyp	Ref	Expression
1		sylsyld.2	. 2 ⊢ (φ → (χ → θ))
2		sylsyld.1	. . 3 ⊢ (φ → ψ)
3		sylsyld.3	. . 3 ⊢ (ψ → (θ → τ))
4	2, 3	syl 15	. 2 ⊢ (φ → (θ → τ))
5	1, 4	syld 40	1 ⊢ (φ → (χ → τ))

Syntax hints:   → wi 4

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7

This theorem is referenced by:  ee02  1377  ax10o  1952  a16gALT  2049  a16g-o  2186  ax10o-o  2203