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Theorem cbval2vw 1944
Description: Rule used to change bound variables, using implicit substitution. (Contributed by NM, 4-Feb-2005.) (Revised by GG, 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 1942 . 2 (𝑥 = 𝑧 → (∀𝑦𝜑 ↔ ∀𝑤𝜓))
32cbvalvw 1931 1 (∀𝑥𝑦𝜑 ↔ ∀𝑧𝑤𝜓)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wb 105  wal 1362
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1458  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546
This theorem depends on definitions:  df-bi 117  df-nf 1472
This theorem is referenced by:  seqf1og  10582
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