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Theorem cbval2v 1798
 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 1421 . 2 𝑧𝜑
2 nfv 1421 . 2 𝑤𝜑
3 nfv 1421 . 2 𝑥𝜓
4 nfv 1421 . 2 𝑦𝜓
5 cbval2v.1 . 2 ((𝑥 = 𝑧𝑦 = 𝑤) → (𝜑𝜓))
61, 2, 3, 4, 5cbval2 1796 1 (∀𝑥𝑦𝜑 ↔ ∀𝑧𝑤𝜓)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98  ∀wal 1241 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428 This theorem depends on definitions:  df-bi 110  df-nf 1350 This theorem is referenced by: (None)
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