Theorem 3anim123i 1211

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 1045	. 2 ⊢ ((𝜑 ∧ 𝜒 ∧ 𝜏) → 𝜓)
3		3anim123i.2	. . 3 ⊢ (𝜒 → 𝜃)
4	3	3ad2ant2 1046	. 2 ⊢ ((𝜑 ∧ 𝜒 ∧ 𝜏) → 𝜃)
5		3anim123i.3	. . 3 ⊢ (𝜏 → 𝜂)
6	5	3ad2ant3 1047	. 2 ⊢ ((𝜑 ∧ 𝜒 ∧ 𝜏) → 𝜂)
7	2, 4, 6	3jca 1204	1 ⊢ ((𝜑 ∧ 𝜒 ∧ 𝜏) → (𝜓 ∧ 𝜃 ∧ 𝜂))

Syntax hints:   → wi 4   ∧ w3a 1005

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 1007

This theorem is referenced by:  3anim1i  1212  3anim2i  1213  3anim3i  1214  syl3an  1316  syl3anl  1325  spc3egv  2899  spc3gv  2900  eloprabga  6118  le2tri3i  8347  fzmmmeqm  10355  elfz1b  10387  elfz0fzfz0  10423  elfzmlbp  10429  elfzo1  10493  flltdivnn0lt  10627  pfxeq  11343  swrdswrd  11352  swrdccat  11382  modmulconst  12464  nndvdslegcd  12616  lgsmulsqcoprm  15865