Theorem jca31 520

Description: Join three consequents. (Contributed by Jeff Hankins, 1-Aug-2009.)

Hypotheses

Ref	Expression
jca31.1	⊢ (φ → ψ)
jca31.2	⊢ (φ → χ)
jca31.3	⊢ (φ → θ)

Assertion

Ref	Expression
jca31	⊢ (φ → ((ψ ∧ χ) ∧ θ))

Proof of Theorem jca31

Step	Hyp	Ref	Expression
1		jca31.1	. . 3 ⊢ (φ → ψ)
2		jca31.2	. . 3 ⊢ (φ → χ)
3	1, 2	jca 518	. 2 ⊢ (φ → (ψ ∧ χ))
4		jca31.3	. 2 ⊢ (φ → θ)
5	3, 4	jca 518	1 ⊢ (φ → ((ψ ∧ χ) ∧ θ))

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:  3jca  1132  syl21anc  1181  f1oiso2  5501