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Theorem rexxfr2d 5411
Description: Transfer existential quantification from a variable 𝑥 to another variable 𝑦 contained in expression 𝐴. (Contributed by Mario Carneiro, 20-Aug-2014.) (Proof shortened by Mario Carneiro, 19-Nov-2016.)
Hypotheses
Ref Expression
ralxfr2d.1 ((𝜑𝑦𝐶) → 𝐴𝑉)
ralxfr2d.2 (𝜑 → (𝑥𝐵 ↔ ∃𝑦𝐶 𝑥 = 𝐴))
ralxfr2d.3 ((𝜑𝑥 = 𝐴) → (𝜓𝜒))
Assertion
Ref Expression
rexxfr2d (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)   𝑉(𝑥,𝑦)

Proof of Theorem rexxfr2d
StepHypRef Expression
1 ralxfr2d.1 . . . 4 ((𝜑𝑦𝐶) → 𝐴𝑉)
2 ralxfr2d.2 . . . 4 (𝜑 → (𝑥𝐵 ↔ ∃𝑦𝐶 𝑥 = 𝐴))
3 ralxfr2d.3 . . . . 5 ((𝜑𝑥 = 𝐴) → (𝜓𝜒))
43notbid 318 . . . 4 ((𝜑𝑥 = 𝐴) → (¬ 𝜓 ↔ ¬ 𝜒))
51, 2, 4ralxfr2d 5410 . . 3 (𝜑 → (∀𝑥𝐵 ¬ 𝜓 ↔ ∀𝑦𝐶 ¬ 𝜒))
65notbid 318 . 2 (𝜑 → (¬ ∀𝑥𝐵 ¬ 𝜓 ↔ ¬ ∀𝑦𝐶 ¬ 𝜒))
7 dfrex2 3073 . 2 (∃𝑥𝐵 𝜓 ↔ ¬ ∀𝑥𝐵 ¬ 𝜓)
8 dfrex2 3073 . 2 (∃𝑦𝐶 𝜒 ↔ ¬ ∀𝑦𝐶 ¬ 𝜒)
96, 7, 83bitr4g 314 1 (𝜑 → (∃𝑥𝐵 𝜓 ↔ ∃𝑦𝐶 𝜒))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395   = wceq 1540  wcel 2108  wral 3061  wrex 3070
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-12 2177  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rex 3071
This theorem is referenced by:  rexrn  7107  reximaOLD  7259  cnpresti  23296  cnprest  23297  1stcrest  23461  subislly  23489  txrest  23639  trfil2  23895  met1stc  24534  metucn  24584  xrlimcnp  27011  esumlub  34061  esumfsup  34071  rexxfr3d  35643  ptrest  37626  djhcvat42  41417
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