Users' Mathboxes Mathbox for Wolf Lammen < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  wl-ax11-lem5 Structured version   Visualization version   GIF version

Theorem wl-ax11-lem5 33496
Description: Lemma. (Contributed by Wolf Lammen, 30-Jun-2019.)
Assertion
Ref Expression
wl-ax11-lem5 (∀𝑢 𝑢 = 𝑦 → (∀𝑢[𝑢 / 𝑦]𝜑 ↔ ∀𝑦𝜑))

Proof of Theorem wl-ax11-lem5
StepHypRef Expression
1 sbequ12r 2150 . . 3 (𝑢 = 𝑦 → ([𝑢 / 𝑦]𝜑𝜑))
21sps 2093 . 2 (∀𝑢 𝑢 = 𝑦 → ([𝑢 / 𝑦]𝜑𝜑))
32dral1 2356 1 (∀𝑢 𝑢 = 𝑦 → (∀𝑢[𝑢 / 𝑦]𝜑 ↔ ∀𝑦𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wal 1521  [wsb 1937
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-10 2059  ax-12 2087  ax-13 2282
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-ex 1745  df-nf 1750  df-sb 1938
This theorem is referenced by:  wl-ax11-lem6  33497
  Copyright terms: Public domain W3C validator