Users' Mathboxes Mathbox for BJ < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  bj-cbval2vv Structured version   Visualization version   GIF version

Theorem bj-cbval2vv 32723
Description: Version of cbval2v 2284 with a dv condition, which does not require ax-13 2245. (Contributed by BJ, 16-Jun-2019.) (Proof modification is discouraged.)
Hypothesis
Ref Expression
bj-cbval2vv.1 ((𝑥 = 𝑧𝑦 = 𝑤) → (𝜑𝜓))
Assertion
Ref Expression
bj-cbval2vv (∀𝑥𝑦𝜑 ↔ ∀𝑧𝑤𝜓)
Distinct variable groups:   𝑧,𝑤,𝜑   𝑥,𝑦,𝜓   𝑥,𝑧,𝑤,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑧,𝑤)

Proof of Theorem bj-cbval2vv
StepHypRef Expression
1 nfv 1842 . 2 𝑧𝜑
2 nfv 1842 . 2 𝑤𝜑
3 nfv 1842 . 2 𝑥𝜓
4 nfv 1842 . 2 𝑦𝜓
5 bj-cbval2vv.1 . 2 ((𝑥 = 𝑧𝑦 = 𝑤) → (𝜑𝜓))
61, 2, 3, 4, 5bj-cbval2v 32721 1 (∀𝑥𝑦𝜑 ↔ ∀𝑧𝑤𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  wal 1480
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-10 2018  ax-11 2033  ax-12 2046
This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1704  df-nf 1709
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator