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Theorem cbvexdva 2433
 Description: Rule used to change the bound variable in an existential quantifier with implicit substitution. Deduction form. Usage of this theorem is discouraged because it depends on ax-13 2392. Use the weaker cbvexdvaw 2047 if possible. (Contributed by David Moews, 1-May-2017.) (New usage is discouraged.)
Hypothesis
Ref Expression
cbvaldva.1 ((𝜑𝑥 = 𝑦) → (𝜓𝜒))
Assertion
Ref Expression
cbvexdva (𝜑 → (∃𝑥𝜓 ↔ ∃𝑦𝜒))
Distinct variable groups:   𝜓,𝑦   𝜒,𝑥   𝜑,𝑥   𝜑,𝑦
Allowed substitution hints:   𝜓(𝑥)   𝜒(𝑦)

Proof of Theorem cbvexdva
StepHypRef Expression
1 nfv 1916 . 2 𝑦𝜑
2 nfvd 1917 . 2 (𝜑 → Ⅎ𝑦𝜓)
3 cbvaldva.1 . . 3 ((𝜑𝑥 = 𝑦) → (𝜓𝜒))
43ex 416 . 2 (𝜑 → (𝑥 = 𝑦 → (𝜓𝜒)))
51, 2, 4cbvexd 2431 1 (𝜑 → (∃𝑥𝜓 ↔ ∃𝑦𝜒))
 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 1912  ax-6 1971  ax-7 2016  ax-11 2162  ax-12 2179  ax-13 2392 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:  cbvex2vv  2438
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