Mathbox for Wolf Lammen < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  wl-rgenw Structured version   Visualization version   GIF version

Theorem wl-rgenw 34711
 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.
StepHypRef Expression
1 wl-dfralsb 34705 . 2 (∀(𝑥 : 𝐴)𝜑 ↔ ∀𝑧(𝑧𝐴 → [𝑧 / 𝑥]𝜑))
2 wl-rgenw.1 . . . 4 𝜑
32sbt 2064 . . 3 [𝑧 / 𝑥]𝜑
43a1i 11 . 2 (𝑧𝐴 → [𝑧 / 𝑥]𝜑)
51, 4mpgbir 1793 1 ∀(𝑥 : 𝐴)𝜑
 Colors of variables: wff setvar class Syntax hints:   → wi 4  [wsb 2062   ∈ wcel 2106  ∀wl-ral 34699 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008 This theorem depends on definitions:  df-bi 208  df-an 397  df-ex 1774  df-sb 2063  df-wl-ral 34704 This theorem is referenced by:  wl-rgen2w  34712
 Copyright terms: Public domain W3C validator