MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  cbvralsvwOLD Structured version   Visualization version   GIF version

Theorem cbvralsvwOLD 3318
Description: Obsolete version of cbvralsvw 3316 as of 21-Aug-2025. (Contributed by NM, 20-Nov-2005.) Avoid ax-13 2376. (Revised by GG, 10-Jan-2024.) (Proof shortened by Wolf Lammen, 8-Mar-2025.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
cbvralsvwOLD (∀𝑥𝐴 𝜑 ↔ ∀𝑦𝐴 [𝑦 / 𝑥]𝜑)
Distinct variable groups:   𝑥,𝑦,𝐴   𝜑,𝑦
Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem cbvralsvwOLD
StepHypRef Expression
1 nfv 1913 . 2 𝑦𝜑
2 nfs1v 2155 . 2 𝑥[𝑦 / 𝑥]𝜑
3 sbequ12 2250 . 2 (𝑥 = 𝑦 → (𝜑 ↔ [𝑦 / 𝑥]𝜑))
41, 2, 3cbvralw 3305 1 (∀𝑥𝐴 𝜑 ↔ ∀𝑦𝐴 [𝑦 / 𝑥]𝜑)
Colors of variables: wff setvar class
Syntax hints:  wb 206  [wsb 2063  wral 3060
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-10 2140  ax-11 2156  ax-12 2176
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-ex 1779  df-nf 1783  df-sb 2064  df-clel 2815  df-nfc 2891  df-ral 3061
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator