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Theorem rexxfr2d 4386
 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 elisset 2700 . . . 4 (𝐴𝑉 → ∃𝑥 𝑥 = 𝐴)
31, 2syl 14 . . 3 ((𝜑𝑦𝐶) → ∃𝑥 𝑥 = 𝐴)
4 ralxfr2d.2 . . . . . . . 8 (𝜑 → (𝑥𝐵 ↔ ∃𝑦𝐶 𝑥 = 𝐴))
54biimprd 157 . . . . . . 7 (𝜑 → (∃𝑦𝐶 𝑥 = 𝐴𝑥𝐵))
6 r19.23v 2541 . . . . . . 7 (∀𝑦𝐶 (𝑥 = 𝐴𝑥𝐵) ↔ (∃𝑦𝐶 𝑥 = 𝐴𝑥𝐵))
75, 6sylibr 133 . . . . . 6 (𝜑 → ∀𝑦𝐶 (𝑥 = 𝐴𝑥𝐵))
87r19.21bi 2520 . . . . 5 ((𝜑𝑦𝐶) → (𝑥 = 𝐴𝑥𝐵))
9 eleq1 2202 . . . . 5 (𝑥 = 𝐴 → (𝑥𝐵𝐴𝐵))
108, 9mpbidi 150 . . . 4 ((𝜑𝑦𝐶) → (𝑥 = 𝐴𝐴𝐵))
1110exlimdv 1791 . . 3 ((𝜑𝑦𝐶) → (∃𝑥 𝑥 = 𝐴𝐴𝐵))
123, 11mpd 13 . 2 ((𝜑𝑦𝐶) → 𝐴𝐵)
134biimpa 294 . 2 ((𝜑𝑥𝐵) → ∃𝑦𝐶 𝑥 = 𝐴)
14 ralxfr2d.3 . 2 ((𝜑𝑥 = 𝐴) → (𝜓𝜒))
1512, 13, 14rexxfrd 4384 1 (𝜑 → (∃𝑥𝐵 𝜓 ↔ ∃𝑦𝐶 𝜒))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   = wceq 1331  ∃wex 1468   ∈ wcel 1480  ∀wral 2416  ∃wrex 2417 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121 This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688 This theorem is referenced by:  rexrn  5557  rexima  5656  cnptopresti  12423  cnptoprest  12424  txrest  12461
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