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Theorem cbvex2vv 2428
Description: Rule used to change bound variables, using implicit substitution. Usage of this theorem is discouraged because it depends on ax-13 2382. Use the weaker cbvex2vw 2048 if possible. (Contributed by NM, 26-Jul-1995.) Remove dependency on ax-10 2143. (Revised by Wolf Lammen, 18-Jul-2021.) (New usage is discouraged.)
Hypothesis
Ref Expression
cbval2vv.1 ((𝑥 = 𝑧𝑦 = 𝑤) → (𝜑𝜓))
Assertion
Ref Expression
cbvex2vv (∃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 cbvex2vv
StepHypRef Expression
1 cbval2vv.1 . . 3 ((𝑥 = 𝑧𝑦 = 𝑤) → (𝜑𝜓))
21cbvexdva 2423 . 2 (𝑥 = 𝑧 → (∃𝑦𝜑 ↔ ∃𝑤𝜓))
32cbvexv 2411 1 (∃𝑥𝑦𝜑 ↔ ∃𝑧𝑤𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  wex 1781
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-11 2159  ax-12 2176  ax-13 2382
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-ex 1782  df-nf 1786
This theorem is referenced by:  cbvex4v  2429
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