Theorem wl-rgenw 34879

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

Hypothesis

Ref	Expression
wl-rgenw.1	⊢ 𝜑

Assertion

Ref	Expression
wl-rgenw	⊢ ∀(𝑥 : 𝐴)𝜑

Proof of Theorem wl-rgenw

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		wl-dfralsb 34873	. 2 ⊢ (∀(𝑥 : 𝐴)𝜑 ↔ ∀𝑧(𝑧 ∈ 𝐴 → [𝑧 / 𝑥]𝜑))
2		wl-rgenw.1	. . . 4 ⊢ 𝜑
3	2	sbt 2070	. . 3 ⊢ [𝑧 / 𝑥]𝜑
4	3	a1i 11	. 2 ⊢ (𝑧 ∈ 𝐴 → [𝑧 / 𝑥]𝜑)
5	1, 4	mpgbir 1799	1 ⊢ ∀(𝑥 : 𝐴)𝜑

Syntax hints:   → wi 4  [wsb 2068   ∈ wcel 2113  ∀wl-ral 34867

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1969  ax-7 2014

This theorem depends on definitions:  df-bi 209  df-an 399  df-ex 1780  df-sb 2069  df-wl-ral 34872

This theorem is referenced by:  wl-rgen2w  34880