Theorem 3anim123i 1184

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 1018	. 2 ⊢ ((𝜑 ∧ 𝜒 ∧ 𝜏) → 𝜓)
3		3anim123i.2	. . 3 ⊢ (𝜒 → 𝜃)
4	3	3ad2ant2 1019	. 2 ⊢ ((𝜑 ∧ 𝜒 ∧ 𝜏) → 𝜃)
5		3anim123i.3	. . 3 ⊢ (𝜏 → 𝜂)
6	5	3ad2ant3 1020	. 2 ⊢ ((𝜑 ∧ 𝜒 ∧ 𝜏) → 𝜂)
7	2, 4, 6	3jca 1177	1 ⊢ ((𝜑 ∧ 𝜒 ∧ 𝜏) → (𝜓 ∧ 𝜃 ∧ 𝜂))

Syntax hints:   → wi 4   ∧ w3a 978

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 980

This theorem is referenced by:  3anim1i  1185  3anim2i  1186  3anim3i  1187  syl3an  1280  syl3anl  1289  spc3egv  2830  spc3gv  2831  eloprabga  5962  le2tri3i  8066  fzmmmeqm  10058  elfz1b  10090  elfz0fzfz0  10126  elfzmlbp  10132  elfzo1  10190  flltdivnn0lt  10304  modmulconst  11830  nndvdslegcd  11966  lgsmulsqcoprm  14450