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Theorem rexxfrd2 4811
 Description: Transfer existence from a variable 𝑥 to another variable 𝑦 contained in expression 𝐴. Variant of rexxfrd 4807. (Contributed by Alexander van der Vekens, 25-Apr-2018.)
Hypotheses
Ref Expression
ralxfrd2.1 ((𝜑𝑦𝐶) → 𝐴𝐵)
ralxfrd2.2 ((𝜑𝑥𝐵) → ∃𝑦𝐶 𝑥 = 𝐴)
ralxfrd2.3 ((𝜑𝑦𝐶𝑥 = 𝐴) → (𝜓𝜒))
Assertion
Ref Expression
rexxfrd2 (𝜑 → (∃𝑥𝐵 𝜓 ↔ ∃𝑦𝐶 𝜒))
Distinct variable groups:   𝑥,𝐴   𝑥,𝑦,𝐵   𝑥,𝐶   𝜒,𝑥   𝜑,𝑥,𝑦   𝜓,𝑦
Allowed substitution hints:   𝜓(𝑥)   𝜒(𝑦)   𝐴(𝑦)   𝐶(𝑦)

Proof of Theorem rexxfrd2
StepHypRef Expression
1 ralxfrd2.1 . . . 4 ((𝜑𝑦𝐶) → 𝐴𝐵)
2 ralxfrd2.2 . . . 4 ((𝜑𝑥𝐵) → ∃𝑦𝐶 𝑥 = 𝐴)
3 ralxfrd2.3 . . . . 5 ((𝜑𝑦𝐶𝑥 = 𝐴) → (𝜓𝜒))
43notbid 307 . . . 4 ((𝜑𝑦𝐶𝑥 = 𝐴) → (¬ 𝜓 ↔ ¬ 𝜒))
51, 2, 4ralxfrd2 4810 . . 3 (𝜑 → (∀𝑥𝐵 ¬ 𝜓 ↔ ∀𝑦𝐶 ¬ 𝜒))
65notbid 307 . 2 (𝜑 → (¬ ∀𝑥𝐵 ¬ 𝜓 ↔ ¬ ∀𝑦𝐶 ¬ 𝜒))
7 dfrex2 2979 . 2 (∃𝑥𝐵 𝜓 ↔ ¬ ∀𝑥𝐵 ¬ 𝜓)
8 dfrex2 2979 . 2 (∃𝑦𝐶 𝜒 ↔ ¬ ∀𝑦𝐶 ¬ 𝜒)
96, 7, 83bitr4g 302 1 (𝜑 → (∃𝑥𝐵 𝜓 ↔ ∃𝑦𝐶 𝜒))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  ∀wral 2896  ∃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-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-v 3175 This theorem is referenced by:  cshimadifsn  13426  cshimadifsn0  13427  ntrclsneine0lem  37382
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