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  5874  exmidapne  7327  nnnq0lem1  7513  prmuloc  7633  suplocexprlemex  7789  prsrlem1  7809  apreap  8614  lemulge11  8893  elnnz  9336  supinfneg  9669  infsupneg  9670  leexp1a  10686  faclbnd6  10836  zfz1isolem1  10932  oddpwdclemdc  12341  ennnfonelemf1  12635  grpidinv2  13190  rhmopp  13732  dvdsrzring  14159  cncnp2m  14467  bj-charfun  15453