Theorem hbimg 31715

Description: A more general form of hbim 2127. (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 1722	. 2 ⊢ ∀𝑥(𝜑 → ∀𝑥𝜓)
3		hbg.2	. 2 ⊢ (𝜒 → ∀𝑥𝜃)
4		hbimtg 31712	. 2 ⊢ ((∀𝑥(𝜑 → ∀𝑥𝜓) ∧ (𝜒 → ∀𝑥𝜃)) → ((𝜓 → 𝜒) → ∀𝑥(𝜑 → 𝜃)))
5	2, 3, 4	mp2an 708	1 ⊢ ((𝜓 → 𝜒) → ∀𝑥(𝜑 → 𝜃))

Syntax hints:   → 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-an 386  df-ex 1705

This theorem is referenced by: (None)