Theorem hbnd 2147

Description: Deduction form of bound-variable hypothesis builder hbn 2146. (Contributed by NM, 3-Jan-2002.)

Hypotheses

Ref	Expression
hbnd.1	⊢ (𝜑 → ∀𝑥𝜑)
hbnd.2	⊢ (𝜑 → (𝜓 → ∀𝑥𝜓))

Assertion

Ref	Expression
hbnd	⊢ (𝜑 → (¬ 𝜓 → ∀𝑥 ¬ 𝜓))

Proof of Theorem hbnd

Step	Hyp	Ref	Expression
1		hbnd.1	. . 3 ⊢ (𝜑 → ∀𝑥𝜑)
2		hbnd.2	. . 3 ⊢ (𝜑 → (𝜓 → ∀𝑥𝜓))
3	1, 2	alrimih 1751	. 2 ⊢ (𝜑 → ∀𝑥(𝜓 → ∀𝑥𝜓))
4		hbnt 2144	. 2 ⊢ (∀𝑥(𝜓 → ∀𝑥𝜓) → (¬ 𝜓 → ∀𝑥 ¬ 𝜓))
5	3, 4	syl 17	1 ⊢ (𝜑 → (¬ 𝜓 → ∀𝑥 ¬ 𝜓))

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1481

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-10 2019  ax-12 2047

This theorem depends on definitions:  df-bi 197  df-or 385  df-ex 1705  df-nf 1710

This theorem is referenced by: (None)