Theorem ggen31 45294

Description: gen31 45370 without virtual deductions. (Contributed by Alan Sare, 22-Jul-2012.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypothesis

Ref	Expression
ggen31.1	⊢ (𝜑 → (𝜓 → (𝜒 → 𝜃)))

Assertion

Ref	Expression
ggen31	⊢ (𝜑 → (𝜓 → (𝜒 → ∀𝑥𝜃)))

Distinct variable groups:   𝜒,𝑥   𝜑,𝑥   𝜓,𝑥

Allowed substitution hint:   𝜃(𝑥)

Proof of Theorem ggen31

Step	Hyp	Ref	Expression
1		ggen31.1	. . . 4 ⊢ (𝜑 → (𝜓 → (𝜒 → 𝜃)))
2	1	imp 412	. . 3 ⊢ ((𝜑 ∧ 𝜓) → (𝜒 → 𝜃))
3	2	alrimdv 1962	. 2 ⊢ ((𝜑 ∧ 𝜓) → (𝜒 → ∀𝑥𝜃))
4	3	ex 418	1 ⊢ (𝜑 → (𝜓 → (𝜒 → ∀𝑥𝜃)))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401  ∀wal 1568

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943

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

This theorem is used by:  onfrALTlem2  45295  gen31  45370