Theorem gen11 44614

Description: Virtual deduction generalizing rule for one quantifying variable and one virtual hypothesis. alrimiv 1925 is gen11 44614 without virtual deductions. (Contributed by Alan Sare, 21-Apr-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypothesis

Ref	Expression
gen11.1	⊢ (   𝜑   ▶   𝜓   )

Assertion

Ref	Expression
gen11	⊢ (   𝜑   ▶   ∀𝑥𝜓   )

Distinct variable group:   𝜑,𝑥

Allowed substitution hint:   𝜓(𝑥)

Proof of Theorem gen11

Step	Hyp	Ref	Expression
1		gen11.1	. . . 4 ⊢ (   𝜑   ▶   𝜓   )
2		dfvd1imp 44573	. . . 4 ⊢ ((   𝜑   ▶   𝜓   ) → (𝜑 → 𝜓))
3	1, 2	ax-mp 5	. . 3 ⊢ (𝜑 → 𝜓)
4	3	alrimiv 1925	. 2 ⊢ (𝜑 → ∀𝑥𝜓)
5		dfvd1impr 44574	. 2 ⊢ ((𝜑 → ∀𝑥𝜓) → (   𝜑   ▶   ∀𝑥𝜓   ))
6	4, 5	ax-mp 5	1 ⊢ (   𝜑   ▶   ∀𝑥𝜓   )

Syntax hints:   → wi 4  ∀wal 1535  (   wvd1 44567

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908

This theorem depends on definitions:  df-bi 207  df-vd1 44568

This theorem is referenced by:  trsspwALT  44816  snssiALTVD  44825  sstrALT2VD  44832  elex2VD  44836  elex22VD  44837  tpid3gVD  44840  trsbcVD  44875  sbcssgVD  44881  csbingVD  44882  onfrALTVD  44889  csbsngVD  44891  csbxpgVD  44892  csbrngVD  44894  csbunigVD  44896  csbfv12gALTVD  44897  ax6e2eqVD  44905  ax6e2ndeqVD  44907  sspwimpVD  44917  sspwimpcfVD  44919