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Theorem rexxfr2d 4913
Description: Transfer universal 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 307 . . . 4 ((𝜑𝑥 = 𝐴) → (¬ 𝜓 ↔ ¬ 𝜒))
51, 2, 4ralxfr2d 4912 . . 3 (𝜑 → (∀𝑥𝐵 ¬ 𝜓 ↔ ∀𝑦𝐶 ¬ 𝜒))
65notbid 307 . 2 (𝜑 → (¬ ∀𝑥𝐵 ¬ 𝜓 ↔ ¬ ∀𝑦𝐶 ¬ 𝜒))
7 dfrex2 3025 . 2 (∃𝑥𝐵 𝜓 ↔ ¬ ∀𝑥𝐵 ¬ 𝜓)
8 dfrex2 3025 . 2 (∃𝑦𝐶 𝜒 ↔ ¬ ∀𝑦𝐶 ¬ 𝜒)
96, 7, 83bitr4g 303 1 (𝜑 → (∃𝑥𝐵 𝜓 ↔ ∃𝑦𝐶 𝜒))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 383   = wceq 1523  wcel 2030  wral 2941  wrex 2942
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-v 3233
This theorem is referenced by:  rexrn  6401  rexima  6537  cnpresti  21140  cnprest  21141  1stcrest  21304  subislly  21332  txrest  21482  trfil2  21738  met1stc  22373  metucn  22423  xrlimcnp  24740  esumlub  30250  esumfsup  30260  ptrest  33538  djhcvat42  37021
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