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Theorem rexxfr2d 5278
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 321 . . . 4 ((𝜑𝑥 = 𝐴) → (¬ 𝜓 ↔ ¬ 𝜒))
51, 2, 4ralxfr2d 5277 . . 3 (𝜑 → (∀𝑥𝐵 ¬ 𝜓 ↔ ∀𝑦𝐶 ¬ 𝜒))
65notbid 321 . 2 (𝜑 → (¬ ∀𝑥𝐵 ¬ 𝜓 ↔ ¬ ∀𝑦𝐶 ¬ 𝜒))
7 dfrex2 3152 . 2 (∃𝑥𝐵 𝜓 ↔ ¬ ∀𝑥𝐵 ¬ 𝜓)
8 dfrex2 3152 . 2 (∃𝑦𝐶 𝜒 ↔ ¬ ∀𝑦𝐶 ¬ 𝜒)
96, 7, 83bitr4g 317 1 (𝜑 → (∃𝑥𝐵 𝜓 ↔ ∃𝑦𝐶 𝜒))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 209  wa 399   = wceq 1542  wcel 2114  wral 3053  wrex 3054
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-12 2179  ax-ext 2710
This theorem depends on definitions:  df-bi 210  df-an 400  df-tru 1545  df-ex 1787  df-sb 2075  df-clab 2717  df-cleq 2730  df-clel 2811  df-ral 3058  df-rex 3059
This theorem is referenced by:  rexrn  6863  rexima  7010  cnpresti  22039  cnprest  22040  1stcrest  22204  subislly  22232  txrest  22382  trfil2  22638  met1stc  23274  metucn  23324  xrlimcnp  25706  esumlub  31598  esumfsup  31608  ptrest  35399  djhcvat42  39052
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