Users' Mathboxes Mathbox for BJ < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  bj-cbvalvv Structured version   Visualization version   GIF version

Theorem bj-cbvalvv 33063
Description: Version of cbvalv 2433 with a dv condition, which does not require ax-13 2407. UPDATE: this is cbvalvw 2124 (which is proved with fewer axioms). (Contributed by BJ, 31-May-2019.) (Proof modification is discouraged.)
Hypothesis
Ref Expression
bj-cbvalvv.1 (𝑥 = 𝑦 → (𝜑𝜓))
Assertion
Ref Expression
bj-cbvalvv (∀𝑥𝜑 ↔ ∀𝑦𝜓)
Distinct variable groups:   𝑥,𝑦   𝜑,𝑦   𝜓,𝑥
Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑦)

Proof of Theorem bj-cbvalvv
StepHypRef Expression
1 nfv 1994 . 2 𝑦𝜑
2 nfv 1994 . 2 𝑥𝜓
3 bj-cbvalvv.1 . 2 (𝑥 = 𝑦 → (𝜑𝜓))
41, 2, 3cbvalv1 2335 1 (∀𝑥𝜑 ↔ ∀𝑦𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wal 1628
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-10 2173  ax-11 2189  ax-12 2202
This theorem depends on definitions:  df-bi 197  df-an 383  df-ex 1852  df-nf 1857
This theorem is referenced by:  bj-zfpow  33125  bj-nfcjust  33176
  Copyright terms: Public domain W3C validator