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  5895  exmidapne  7371  nnnq0lem1  7558  prmuloc  7678  suplocexprlemex  7834  prsrlem1  7854  apreap  8659  lemulge11  8938  elnnz  9381  supinfneg  9715  infsupneg  9716  leexp1a  10737  faclbnd6  10887  zfz1isolem1  10983  oddpwdclemdc  12437  ennnfonelemf1  12731  grpidinv2  13332  rhmopp  13880  dvdsrzring  14307  cncnp2m  14645  bj-charfun  15676