Theorem jca32 521

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

Hypotheses

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

Assertion

Ref	Expression
jca32	⊢ (φ → (ψ ∧ (χ ∧ θ)))

Proof of Theorem jca32

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

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:  syl12anc  1180  euan  2261