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Theorem cbval2vw 2044
Description: Rule used to change bound variables, using implicit substitution. Version of cbval2vv 2412 with more disjoint variable conditions, which requires fewer axioms . (Contributed by NM, 4-Feb-2005.) Avoid ax-13 2371. (Revised by Gino Giotto, 10-Jan-2024.)
Hypothesis
Ref Expression
cbval2vw.1 ((𝑥 = 𝑧𝑦 = 𝑤) → (𝜑𝜓))
Assertion
Ref Expression
cbval2vw (∀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 cbval2vw
StepHypRef Expression
1 cbval2vw.1 . . 3 ((𝑥 = 𝑧𝑦 = 𝑤) → (𝜑𝜓))
21cbvaldvaw 2042 . 2 (𝑥 = 𝑧 → (∀𝑦𝜑 ↔ ∀𝑤𝜓))
32cbvalvw 2040 1 (∀𝑥𝑦𝜑 ↔ ∀𝑧𝑤𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397  wal 1540
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012
This theorem depends on definitions:  df-bi 206  df-an 398  df-ex 1783
This theorem is referenced by:  seqf1o  13955  fi1uzind  14402  brfi1indALT  14405  mbfresfi  36170
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