Theorem gen22 42242

Description: Virtual deduction generalizing rule for two quantifying variables and two virtual hypothesis. (Contributed by Alan Sare, 25-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypothesis

Ref	Expression
gen22.1	⊢ (   𝜑   ,   𝜓   ▶   𝜒   )

Assertion

Ref	Expression
gen22	⊢ (   𝜑   ,   𝜓   ▶   ∀𝑥∀𝑦𝜒   )

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

Allowed substitution hints:   𝜒(𝑥,𝑦)

Proof of Theorem gen22

Step	Hyp	Ref	Expression
1		gen22.1	. . . . 5 ⊢ (   𝜑   ,   𝜓   ▶   𝜒   )
2	1	dfvd2i 42205	. . . 4 ⊢ (𝜑 → (𝜓 → 𝜒))
3	2	alrimdv 1932	. . 3 ⊢ (𝜑 → (𝜓 → ∀𝑦𝜒))
4	3	alrimdv 1932	. 2 ⊢ (𝜑 → (𝜓 → ∀𝑥∀𝑦𝜒))
5	4	dfvd2ir 42206	1 ⊢ (   𝜑   ,   𝜓   ▶   ∀𝑥∀𝑦𝜒   )

Syntax hints:  ∀wal 1537  (   wvd2 42197

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913

This theorem depends on definitions:  df-bi 206  df-an 397  df-vd2 42198

This theorem is referenced by: (None)