Theorem 3anim123i 1186

Description: Join antecedents and consequents with conjunction. (Contributed by NM, 8-Apr-1994.)

Hypotheses

Ref	Expression
3anim123i.1	⊢ (𝜑 → 𝜓)
3anim123i.2	⊢ (𝜒 → 𝜃)
3anim123i.3	⊢ (𝜏 → 𝜂)

Assertion

Ref	Expression
3anim123i	⊢ ((𝜑 ∧ 𝜒 ∧ 𝜏) → (𝜓 ∧ 𝜃 ∧ 𝜂))

Proof of Theorem 3anim123i

Step	Hyp	Ref	Expression
1		3anim123i.1	. . 3 ⊢ (𝜑 → 𝜓)
2	1	3ad2ant1 1020	. 2 ⊢ ((𝜑 ∧ 𝜒 ∧ 𝜏) → 𝜓)
3		3anim123i.2	. . 3 ⊢ (𝜒 → 𝜃)
4	3	3ad2ant2 1021	. 2 ⊢ ((𝜑 ∧ 𝜒 ∧ 𝜏) → 𝜃)
5		3anim123i.3	. . 3 ⊢ (𝜏 → 𝜂)
6	5	3ad2ant3 1022	. 2 ⊢ ((𝜑 ∧ 𝜒 ∧ 𝜏) → 𝜂)
7	2, 4, 6	3jca 1179	1 ⊢ ((𝜑 ∧ 𝜒 ∧ 𝜏) → (𝜓 ∧ 𝜃 ∧ 𝜂))

Syntax hints:   → wi 4   ∧ w3a 980

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108

This theorem depends on definitions:  df-bi 117  df-3an 982

This theorem is referenced by:  3anim1i  1187  3anim2i  1188  3anim3i  1189  syl3an  1291  syl3anl  1300  spc3egv  2844  spc3gv  2845  eloprabga  5979  le2tri3i  8091  fzmmmeqm  10083  elfz1b  10115  elfz0fzfz0  10151  elfzmlbp  10157  elfzo1  10215  flltdivnn0lt  10330  modmulconst  11857  nndvdslegcd  11993  lgsmulsqcoprm  14884