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Theorem cbval2v 1843
Description: Rule used to change bound variables, using implicit substitution. (Contributed by NM, 4-Feb-2005.)
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 nfv 1464 . 2 𝑧𝜑
2 nfv 1464 . 2 𝑤𝜑
3 nfv 1464 . 2 𝑥𝜓
4 nfv 1464 . 2 𝑦𝜓
5 cbval2v.1 . 2 ((𝑥 = 𝑧𝑦 = 𝑤) → (𝜑𝜓))
61, 2, 3, 4, 5cbval2 1841 1 (∀𝑥𝑦𝜑 ↔ ∀𝑧𝑤𝜓)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wb 103  wal 1285
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-4 1443  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471
This theorem depends on definitions:  df-bi 115  df-nf 1393
This theorem is referenced by: (None)
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