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Theorem cbvrexdva2 3152
 Description: Rule used to change the bound variable in a restricted existential quantifier with implicit substitution which also changes the quantifier domain. Deduction form. (Contributed by David Moews, 1-May-2017.)
Hypotheses
Ref Expression
cbvraldva2.1 ((𝜑𝑥 = 𝑦) → (𝜓𝜒))
cbvraldva2.2 ((𝜑𝑥 = 𝑦) → 𝐴 = 𝐵)
Assertion
Ref Expression
cbvrexdva2 (𝜑 → (∃𝑥𝐴 𝜓 ↔ ∃𝑦𝐵 𝜒))
Distinct variable groups:   𝑦,𝐴   𝜓,𝑦   𝑥,𝐵   𝜒,𝑥   𝜑,𝑥,𝑦
Allowed substitution hints:   𝜓(𝑥)   𝜒(𝑦)   𝐴(𝑥)   𝐵(𝑦)

Proof of Theorem cbvrexdva2
StepHypRef Expression
1 simpr 476 . . . . 5 ((𝜑𝑥 = 𝑦) → 𝑥 = 𝑦)
2 cbvraldva2.2 . . . . 5 ((𝜑𝑥 = 𝑦) → 𝐴 = 𝐵)
31, 2eleq12d 2682 . . . 4 ((𝜑𝑥 = 𝑦) → (𝑥𝐴𝑦𝐵))
4 cbvraldva2.1 . . . 4 ((𝜑𝑥 = 𝑦) → (𝜓𝜒))
53, 4anbi12d 743 . . 3 ((𝜑𝑥 = 𝑦) → ((𝑥𝐴𝜓) ↔ (𝑦𝐵𝜒)))
65cbvexdva 2271 . 2 (𝜑 → (∃𝑥(𝑥𝐴𝜓) ↔ ∃𝑦(𝑦𝐵𝜒)))
7 df-rex 2902 . 2 (∃𝑥𝐴 𝜓 ↔ ∃𝑥(𝑥𝐴𝜓))
8 df-rex 2902 . 2 (∃𝑦𝐵 𝜒 ↔ ∃𝑦(𝑦𝐵𝜒))
96, 7, 83bitr4g 302 1 (𝜑 → (∃𝑥𝐴 𝜓 ↔ ∃𝑦𝐵 𝜒))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475  ∃wex 1695   ∈ wcel 1977  ∃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:  cbvrexdva  3154  mreexexlemd  16127  eulerpartlemgvv  29765
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