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Theorem cbval2v 2273
 Description: Rule used to change bound variables, using implicit substitution. (Contributed by NM, 4-Feb-2005.) Remove dependency on ax-10 2006. (Revised by Wolf Lammen, 18-Jul-2021.)
Hypothesis
Ref Expression
cbval2v.1 ((𝑥 = 𝑧𝑦 = 𝑤) → (𝜑𝜓))
Assertion
Ref Expression
cbval2v (∀𝑥𝑦𝜑 ↔ ∀𝑧𝑤𝜓)
Distinct variable groups:   𝑧,𝑤,𝜑   𝑥,𝑦,𝜓   𝑥,𝑤   𝑦,𝑧
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑧,𝑤)

Proof of Theorem cbval2v
StepHypRef Expression
1 cbval2v.1 . . 3 ((𝑥 = 𝑧𝑦 = 𝑤) → (𝜑𝜓))
21cbvaldva 2269 . 2 (𝑥 = 𝑧 → (∀𝑦𝜑 ↔ ∀𝑤𝜓))
32cbvalv 2261 1 (∀𝑥𝑦𝜑 ↔ ∀𝑧𝑤𝜓)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383  ∀wal 1473 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234 This theorem depends on definitions:  df-bi 196  df-an 385  df-ex 1696  df-nf 1701 This theorem is referenced by:  seqf1o  12704  fi1uzind  13134  brfi1indALT  13137  fi1uzindOLD  13140  brfi1indALTOLD  13143  mbfresfi  32626
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