Theorem hb3and 1478

Description: Deduction form of bound-variable hypothesis builder hb3an 1538. (Contributed by NM, 17-Feb-2013.)

Hypotheses

Ref	Expression
hb3and.1	⊢ (𝜑 → (𝜓 → ∀𝑥𝜓))
hb3and.2	⊢ (𝜑 → (𝜒 → ∀𝑥𝜒))
hb3and.3	⊢ (𝜑 → (𝜃 → ∀𝑥𝜃))

Assertion

Ref	Expression
hb3and	⊢ (𝜑 → ((𝜓 ∧ 𝜒 ∧ 𝜃) → ∀𝑥(𝜓 ∧ 𝜒 ∧ 𝜃)))

Proof of Theorem hb3and

Step	Hyp	Ref	Expression
1		hb3and.1	. . 3 ⊢ (𝜑 → (𝜓 → ∀𝑥𝜓))
2		hb3and.2	. . 3 ⊢ (𝜑 → (𝜒 → ∀𝑥𝜒))
3		hb3and.3	. . 3 ⊢ (𝜑 → (𝜃 → ∀𝑥𝜃))
4	1, 2, 3	3anim123d 1309	. 2 ⊢ (𝜑 → ((𝜓 ∧ 𝜒 ∧ 𝜃) → (∀𝑥𝜓 ∧ ∀𝑥𝜒 ∧ ∀𝑥𝜃)))
5		19.26-3an 1471	. 2 ⊢ (∀𝑥(𝜓 ∧ 𝜒 ∧ 𝜃) ↔ (∀𝑥𝜓 ∧ ∀𝑥𝜒 ∧ ∀𝑥𝜃))
6	4, 5	syl6ibr 161	1 ⊢ (𝜑 → ((𝜓 ∧ 𝜒 ∧ 𝜃) → ∀𝑥(𝜓 ∧ 𝜒 ∧ 𝜃)))

Syntax hints:   → wi 4   ∧ w3a 968  ∀wal 1341

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1435  ax-gen 1437

This theorem depends on definitions:  df-bi 116  df-3an 970

This theorem is referenced by: (None)