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Theorem cbvrexdva 3154
 Description: Rule used to change the bound variable in a restricted existential quantifier with implicit substitution. Deduction form. (Contributed by David Moews, 1-May-2017.)
Hypothesis
Ref Expression
cbvraldva.1 ((𝜑𝑥 = 𝑦) → (𝜓𝜒))
Assertion
Ref Expression
cbvrexdva (𝜑 → (∃𝑥𝐴 𝜓 ↔ ∃𝑦𝐴 𝜒))
Distinct variable groups:   𝜓,𝑦   𝜒,𝑥   𝑥,𝐴,𝑦   𝜑,𝑥,𝑦
Allowed substitution hints:   𝜓(𝑥)   𝜒(𝑦)

Proof of Theorem cbvrexdva
StepHypRef Expression
1 cbvraldva.1 . 2 ((𝜑𝑥 = 𝑦) → (𝜓𝜒))
2 eqidd 2611 . 2 ((𝜑𝑥 = 𝑦) → 𝐴 = 𝐴)
31, 2cbvrexdva2 3152 1 (𝜑 → (∃𝑥𝐴 𝜓 ↔ ∃𝑦𝐴 𝜒))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383  ∃wrex 2897 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-an 385  df-ex 1696  df-nf 1701  df-cleq 2603  df-clel 2606  df-rex 2902 This theorem is referenced by:  tfrlem3a  7360  trgcopy  25496  trgcopyeu  25498  acopyeu  25525  tgasa1  25539  2sqmo  28980  dispcmp  29254  f1omptsn  32360
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