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Theorem cbvex2vw 2036
Description: Rule used to change bound variables, using implicit substitution. Version of cbvex2vv 2407 with more disjoint variable conditions, which requires fewer axioms . (Contributed by NM, 26-Jul-1995.) Avoid ax-13 2365. (Revised by Gino Giotto, 10-Jan-2024.)
Hypothesis
Ref Expression
cbval2vw.1 ((𝑥 = 𝑧𝑦 = 𝑤) → (𝜑𝜓))
Assertion
Ref Expression
cbvex2vw (∃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 cbvex2vw
StepHypRef Expression
1 cbval2vw.1 . . 3 ((𝑥 = 𝑧𝑦 = 𝑤) → (𝜑𝜓))
21cbvexdvaw 2034 . 2 (𝑥 = 𝑧 → (∃𝑦𝜑 ↔ ∃𝑤𝜓))
32cbvexvw 2032 1 (∃𝑥𝑦𝜑 ↔ ∃𝑧𝑤𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395  wex 1773
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003
This theorem depends on definitions:  df-bi 206  df-an 396  df-ex 1774
This theorem is referenced by:  cbvex4vw  2037  cbvopabv  5214  bj-cbvex4vv  36191  funop1  46560  uspgrsprf1  47094
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