Theorem hband 1418

Description: Deduction form of bound-variable hypothesis builder hban 1479. (Contributed by NM, 2-Jan-2002.)

Hypotheses

Ref	Expression
hband.1	⊢ (𝜑 → (𝜓 → ∀𝑥𝜓))
hband.2	⊢ (𝜑 → (𝜒 → ∀𝑥𝜒))

Assertion

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

Proof of Theorem hband

Step	Hyp	Ref	Expression
1		hband.1	. . 3 ⊢ (𝜑 → (𝜓 → ∀𝑥𝜓))
2		hband.2	. . 3 ⊢ (𝜑 → (𝜒 → ∀𝑥𝜒))
3	1, 2	anim12d 328	. 2 ⊢ (𝜑 → ((𝜓 ∧ 𝜒) → (∀𝑥𝜓 ∧ ∀𝑥𝜒)))
4		19.26 1410	. 2 ⊢ (∀𝑥(𝜓 ∧ 𝜒) ↔ (∀𝑥𝜓 ∧ ∀𝑥𝜒))
5	3, 4	syl6ibr 160	1 ⊢ (𝜑 → ((𝜓 ∧ 𝜒) → ∀𝑥(𝜓 ∧ 𝜒)))

Syntax hints:   → wi 4   ∧ wa 102  ∀wal 1282

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1376  ax-gen 1378

This theorem depends on definitions:  df-bi 115

This theorem is referenced by: (None)