Theorem hband 1465

Description: Deduction form of bound-variable hypothesis builder hban 1526. (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 333	. 2 ⊢ (𝜑 → ((𝜓 ∧ 𝜒) → (∀𝑥𝜓 ∧ ∀𝑥𝜒)))
4		19.26 1457	. 2 ⊢ (∀𝑥(𝜓 ∧ 𝜒) ↔ (∀𝑥𝜓 ∧ ∀𝑥𝜒))
5	3, 4	syl6ibr 161	1 ⊢ (𝜑 → ((𝜓 ∧ 𝜒) → ∀𝑥(𝜓 ∧ 𝜒)))

Syntax hints:   → wi 4   ∧ wa 103  ∀wal 1329

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 1423  ax-gen 1425

This theorem depends on definitions:  df-bi 116

This theorem is referenced by: (None)