Theorem wl-rgen 34394

Description: Generalization rule for restricted quantification. (Contributed by Wolf Lammen, 10-Jun-2023.)

Hypothesis

Ref	Expression
wl-rgen.1	⊢ (𝑥 ∈ 𝐴 → 𝜑)

Assertion

Ref	Expression
wl-rgen	⊢ ∀(𝑥 : 𝐴)𝜑

Distinct variable group:   𝑥,𝐴

Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem wl-rgen

Step	Hyp	Ref	Expression
1		wl-dfralv 34393	. 2 ⊢ (∀(𝑥 : 𝐴)𝜑 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝜑))
2		wl-rgen.1	. 2 ⊢ (𝑥 ∈ 𝐴 → 𝜑)
3	1, 2	mpgbir 1785	1 ⊢ ∀(𝑥 : 𝐴)𝜑

Syntax hints:   → wi 4   ∈ wcel 2083  ∀wl-ral 34383

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1781  ax-4 1795  ax-5 1892  ax-6 1951  ax-7 1996  ax-8 2085  ax-11 2128

This theorem depends on definitions:  df-bi 208  df-an 397  df-ex 1766  df-clel 2865  df-wl-ral 34388

This theorem is referenced by:  wl-ralel  34397