Theorem hbimg 36571

Description: A more general form of hbim 2332. (Contributed by Scott Fenton, 13-Dec-2010.)

Hypotheses

Ref	Expression
hbg.1	⊢ (𝜑 → ∀𝑥𝜓)
hbg.2	⊢ (𝜒 → ∀𝑥𝜃)

Assertion

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

Proof of Theorem hbimg

Step	Hyp	Ref	Expression
1		hbg.1	. . 3 ⊢ (𝜑 → ∀𝑥𝜓)
2	1	ax-gen 1827	. 2 ⊢ ∀𝑥(𝜑 → ∀𝑥𝜓)
3		hbg.2	. 2 ⊢ (𝜒 → ∀𝑥𝜃)
4		hbimtg 36568	. 2 ⊢ ((∀𝑥(𝜑 → ∀𝑥𝜓) ∧ (𝜒 → ∀𝑥𝜃)) → ((𝜓 → 𝜒) → ∀𝑥(𝜑 → 𝜃)))
5	2, 3, 4	mp2an 705	1 ⊢ ((𝜓 → 𝜒) → ∀𝑥(𝜑 → 𝜃))

This proof depends on syntax axioms:   → wi 4  ∀wal 1567

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-10 2177  ax-12 2212

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1812

This theorem is used by: (None)