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Theorem cbvrexdva 3445
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 2825 . 2 ((𝜑𝑥 = 𝑦) → 𝐴 = 𝐴)
31, 2cbvrexdva2 3442 1 (𝜑 → (∃𝑥𝐴 𝜓 ↔ ∃𝑦𝐴 𝜒))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  wrex 3133
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-8 2117  ax-9 2125  ax-ext 2796
This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-cleq 2817  df-clel 2896  df-rex 3138
This theorem is referenced by:  tfrlem3a  7996  2sqmo  26010  trgcopy  26587  trgcopyeu  26589  acopyeu  26617  tgasa1  26641  dispcmp  31144  satffunlem1lem1  32667  satffunlem2lem1  32669  f1omptsn  34656  pibt2  34736  prjsprel  39430
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