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Theorem rexxfr 4318
 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 ralxfr.1 . . . 4 (𝑦𝐶𝐴𝐵)
21adantl 272 . . 3 ((⊤ ∧ 𝑦𝐶) → 𝐴𝐵)
3 ralxfr.2 . . . 4 (𝑥𝐵 → ∃𝑦𝐶 𝑥 = 𝐴)
43adantl 272 . . 3 ((⊤ ∧ 𝑥𝐵) → ∃𝑦𝐶 𝑥 = 𝐴)
5 ralxfr.3 . . . 4 (𝑥 = 𝐴 → (𝜑𝜓))
65adantl 272 . . 3 ((⊤ ∧ 𝑥 = 𝐴) → (𝜑𝜓))
72, 4, 6rexxfrd 4313 . 2 (⊤ → (∃𝑥𝐵 𝜑 ↔ ∃𝑦𝐶 𝜓))
87mptru 1305 1 (∃𝑥𝐵 𝜑 ↔ ∃𝑦𝐶 𝜓)
 Colors of variables: wff set class Syntax hints:   → wi 4   ↔ wb 104   = wceq 1296  ⊤wtru 1297   ∈ wcel 1445  ∃wrex 2371 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077 This theorem depends on definitions:  df-bi 116  df-tru 1299  df-nf 1402  df-sb 1700  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-ral 2375  df-rex 2376  df-v 2635 This theorem is referenced by: (None)
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