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Theorem rexxfrd 4195
 Description: Transfer universal quantification from a variable 𝑥 to another variable 𝑦 contained in expression 𝐴. (Contributed by FL, 10-Apr-2007.) (Revised by Mario Carneiro, 15-Aug-2014.)
Hypotheses
Ref Expression
ralxfrd.1 ((𝜑𝑦𝐶) → 𝐴𝐵)
ralxfrd.2 ((𝜑𝑥𝐵) → ∃𝑦𝐶 𝑥 = 𝐴)
ralxfrd.3 ((𝜑𝑥 = 𝐴) → (𝜓𝜒))
Assertion
Ref Expression
rexxfrd (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)

Proof of Theorem rexxfrd
StepHypRef Expression
1 nfv 1421 . . . . 5 𝑦𝜓
2119.3 1446 . . . 4 (∀𝑦𝜓𝜓)
3 ralxfrd.2 . . . . 5 ((𝜑𝑥𝐵) → ∃𝑦𝐶 𝑥 = 𝐴)
4 df-rex 2312 . . . . . . . 8 (∃𝑦𝐶 𝑥 = 𝐴 ↔ ∃𝑦(𝑦𝐶𝑥 = 𝐴))
5 19.29 1511 . . . . . . . . . 10 ((∀𝑦𝜓 ∧ ∃𝑦(𝑦𝐶𝑥 = 𝐴)) → ∃𝑦(𝜓 ∧ (𝑦𝐶𝑥 = 𝐴)))
6 an12 495 . . . . . . . . . . 11 ((𝜓 ∧ (𝑦𝐶𝑥 = 𝐴)) ↔ (𝑦𝐶 ∧ (𝜓𝑥 = 𝐴)))
76exbii 1496 . . . . . . . . . 10 (∃𝑦(𝜓 ∧ (𝑦𝐶𝑥 = 𝐴)) ↔ ∃𝑦(𝑦𝐶 ∧ (𝜓𝑥 = 𝐴)))
85, 7sylib 127 . . . . . . . . 9 ((∀𝑦𝜓 ∧ ∃𝑦(𝑦𝐶𝑥 = 𝐴)) → ∃𝑦(𝑦𝐶 ∧ (𝜓𝑥 = 𝐴)))
9 df-rex 2312 . . . . . . . . 9 (∃𝑦𝐶 (𝜓𝑥 = 𝐴) ↔ ∃𝑦(𝑦𝐶 ∧ (𝜓𝑥 = 𝐴)))
108, 9sylibr 137 . . . . . . . 8 ((∀𝑦𝜓 ∧ ∃𝑦(𝑦𝐶𝑥 = 𝐴)) → ∃𝑦𝐶 (𝜓𝑥 = 𝐴))
114, 10sylan2b 271 . . . . . . 7 ((∀𝑦𝜓 ∧ ∃𝑦𝐶 𝑥 = 𝐴) → ∃𝑦𝐶 (𝜓𝑥 = 𝐴))
12 ralxfrd.3 . . . . . . . . . . 11 ((𝜑𝑥 = 𝐴) → (𝜓𝜒))
1312biimpd 132 . . . . . . . . . 10 ((𝜑𝑥 = 𝐴) → (𝜓𝜒))
1413expimpd 345 . . . . . . . . 9 (𝜑 → ((𝑥 = 𝐴𝜓) → 𝜒))
1514ancomsd 256 . . . . . . . 8 (𝜑 → ((𝜓𝑥 = 𝐴) → 𝜒))
1615reximdv 2420 . . . . . . 7 (𝜑 → (∃𝑦𝐶 (𝜓𝑥 = 𝐴) → ∃𝑦𝐶 𝜒))
1711, 16syl5 28 . . . . . 6 (𝜑 → ((∀𝑦𝜓 ∧ ∃𝑦𝐶 𝑥 = 𝐴) → ∃𝑦𝐶 𝜒))
1817adantr 261 . . . . 5 ((𝜑𝑥𝐵) → ((∀𝑦𝜓 ∧ ∃𝑦𝐶 𝑥 = 𝐴) → ∃𝑦𝐶 𝜒))
193, 18mpan2d 404 . . . 4 ((𝜑𝑥𝐵) → (∀𝑦𝜓 → ∃𝑦𝐶 𝜒))
202, 19syl5bir 142 . . 3 ((𝜑𝑥𝐵) → (𝜓 → ∃𝑦𝐶 𝜒))
2120rexlimdva 2433 . 2 (𝜑 → (∃𝑥𝐵 𝜓 → ∃𝑦𝐶 𝜒))
22 ralxfrd.1 . . . 4 ((𝜑𝑦𝐶) → 𝐴𝐵)
2312adantlr 446 . . . 4 (((𝜑𝑦𝐶) ∧ 𝑥 = 𝐴) → (𝜓𝜒))
2422, 23rspcedv 2660 . . 3 ((𝜑𝑦𝐶) → (𝜒 → ∃𝑥𝐵 𝜓))
2524rexlimdva 2433 . 2 (𝜑 → (∃𝑦𝐶 𝜒 → ∃𝑥𝐵 𝜓))
2621, 25impbid 120 1 (𝜑 → (∃𝑥𝐵 𝜓 ↔ ∃𝑦𝐶 𝜒))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98  ∀wal 1241   = wceq 1243  ∃wex 1381   ∈ wcel 1393  ∃wrex 2307 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022 This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559 This theorem is referenced by:  rexxfr2d  4197  rexxfr  4200
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