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Theorem rexxfr 5086
Description: Transfer existence from a variable 𝑥 to another variable 𝑦 contained in expression 𝐴. (Contributed by NM, 10-Jun-2005.) (Revised by Mario Carneiro, 15-Aug-2014.)
Hypotheses
Ref Expression
ralxfr.1 (𝑦𝐶𝐴𝐵)
ralxfr.2 (𝑥𝐵 → ∃𝑦𝐶 𝑥 = 𝐴)
ralxfr.3 (𝑥 = 𝐴 → (𝜑𝜓))
Assertion
Ref Expression
rexxfr (∃𝑥𝐵 𝜑 ↔ ∃𝑦𝐶 𝜓)
Distinct variable groups:   𝜓,𝑥   𝜑,𝑦   𝑥,𝐴   𝑥,𝑦,𝐵   𝑥,𝐶
Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑦)   𝐴(𝑦)   𝐶(𝑦)

Proof of Theorem rexxfr
StepHypRef Expression
1 dfrex2 3176 . 2 (∃𝑥𝐵 𝜑 ↔ ¬ ∀𝑥𝐵 ¬ 𝜑)
2 dfrex2 3176 . . 3 (∃𝑦𝐶 𝜓 ↔ ¬ ∀𝑦𝐶 ¬ 𝜓)
3 ralxfr.1 . . . 4 (𝑦𝐶𝐴𝐵)
4 ralxfr.2 . . . 4 (𝑥𝐵 → ∃𝑦𝐶 𝑥 = 𝐴)
5 ralxfr.3 . . . . 5 (𝑥 = 𝐴 → (𝜑𝜓))
65notbid 310 . . . 4 (𝑥 = 𝐴 → (¬ 𝜑 ↔ ¬ 𝜓))
73, 4, 6ralxfr 5084 . . 3 (∀𝑥𝐵 ¬ 𝜑 ↔ ∀𝑦𝐶 ¬ 𝜓)
82, 7xchbinxr 327 . 2 (∃𝑦𝐶 𝜓 ↔ ¬ ∀𝑥𝐵 ¬ 𝜑)
91, 8bitr4i 270 1 (∃𝑥𝐵 𝜑 ↔ ∃𝑦𝐶 𝜓)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 198   = wceq 1653  wcel 2157  wral 3089  wrex 3090
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-ext 2777
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ral 3094  df-rex 3095  df-v 3387
This theorem is referenced by:  infm3  11274  reeff1o  24542  moxfr  38037
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