Theorem 3anim123i 1137

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 976	. 2 ⊢ ((φ ∧ χ ∧ τ) → ψ)
3		3anim123i.2	. . 3 ⊢ (χ → θ)
4	3	3ad2ant2 977	. 2 ⊢ ((φ ∧ χ ∧ τ) → θ)
5		3anim123i.3	. . 3 ⊢ (τ → η)
6	5	3ad2ant3 978	. 2 ⊢ ((φ ∧ χ ∧ τ) → η)
7	2, 4, 6	3jca 1132	1 ⊢ ((φ ∧ χ ∧ τ) → (ψ ∧ θ ∧ η))

This proof depends on syntax axioms:   → wi 4   ∧ w3a 934

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This proof depends on definitions:  df-bi 177  df-an 360  df-3an 936

This theorem is used by:  3anim1i  1138  3anim3i  1139  syl3an  1224  syl3anl  1233  spc3egv  2944