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Theorem cbvex2v 2421
Description: Rule used to change bound variables, using implicit substitution. (Contributed by NM, 26-Jul-1995.) Remove dependency on ax-10 2185. (Revised by Wolf Lammen, 18-Jul-2021.)
Hypothesis
Ref Expression
cbval2v.1 ((𝑥 = 𝑧𝑦 = 𝑤) → (𝜑𝜓))
Assertion
Ref Expression
cbvex2v (∃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 cbvex2v
StepHypRef Expression
1 cbval2v.1 . . 3 ((𝑥 = 𝑧𝑦 = 𝑤) → (𝜑𝜓))
21cbvexdva 2419 . 2 (𝑥 = 𝑧 → (∃𝑦𝜑 ↔ ∃𝑤𝜓))
32cbvexv 2413 1 (∃𝑥𝑦𝜑 ↔ ∃𝑧𝑤𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 385  wex 1875
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-11 2200  ax-12 2213  ax-13 2377
This theorem depends on definitions:  df-bi 199  df-an 386  df-ex 1876  df-nf 1880
This theorem is referenced by:  cbvex4v  2422  funop1  42138  uspgrsprf1  42554
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