Theorem 3jcad 1085

Description: Deduction conjoining the consequents of three implications. (Contributed by NM, 25-Sep-2005.)

Hypotheses

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

Assertion

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

Proof of Theorem 3jcad

Step	Hyp	Ref	Expression
1		3jcad.1	. . . 4 ⊢ (𝜑 → (𝜓 → 𝜒))
2	1	imp 115	. . 3 ⊢ ((𝜑 ∧ 𝜓) → 𝜒)
3		3jcad.2	. . . 4 ⊢ (𝜑 → (𝜓 → 𝜃))
4	3	imp 115	. . 3 ⊢ ((𝜑 ∧ 𝜓) → 𝜃)
5		3jcad.3	. . . 4 ⊢ (𝜑 → (𝜓 → 𝜏))
6	5	imp 115	. . 3 ⊢ ((𝜑 ∧ 𝜓) → 𝜏)
7	2, 4, 6	3jca 1084	. 2 ⊢ ((𝜑 ∧ 𝜓) → (𝜒 ∧ 𝜃 ∧ 𝜏))
8	7	ex 108	1 ⊢ (𝜑 → (𝜓 → (𝜒 ∧ 𝜃 ∧ 𝜏)))

Syntax hints:   → wi 4   ∧ wa 97   ∧ w3a 885

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101

This theorem depends on definitions:  df-bi 110  df-3an 887

This theorem is referenced by:  ixxssixx  8771  iccid  8794  fzen  8907