Theorem jca31 309

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 306	. 2 ⊢ (𝜑 → (𝜓 ∧ 𝜒))
4		jca31.3	. 2 ⊢ (𝜑 → 𝜃)
5	3, 4	jca 306	1 ⊢ (𝜑 → ((𝜓 ∧ 𝜒) ∧ 𝜃))

Syntax hints:   → wi 4   ∧ wa 104

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia3 108

This theorem is referenced by:  3jca  1179  syl21anbrc  1184  syl21anc  1248  f1oiso2  5844  exmidapne  7277  nnnq0lem1  7463  prmuloc  7583  suplocexprlemex  7739  prsrlem1  7759  apreap  8562  lemulge11  8841  elnnz  9281  supinfneg  9613  infsupneg  9614  leexp1a  10593  faclbnd6  10742  zfz1isolem1  10838  oddpwdclemdc  12191  ennnfonelemf1  12437  grpidinv2  12968  rhmopp  13487  dvdsrzring  13863  cncnp2m  14115  bj-charfun  14943