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Theorem rexxfrd 4352
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 1491 . . . . 5 𝑦𝜓
2119.3 1516 . . . 4 (∀𝑦𝜓𝜓)
3 ralxfrd.2 . . . . 5 ((𝜑𝑥𝐵) → ∃𝑦𝐶 𝑥 = 𝐴)
4 df-rex 2397 . . . . . . . 8 (∃𝑦𝐶 𝑥 = 𝐴 ↔ ∃𝑦(𝑦𝐶𝑥 = 𝐴))
5 19.29 1582 . . . . . . . . . 10 ((∀𝑦𝜓 ∧ ∃𝑦(𝑦𝐶𝑥 = 𝐴)) → ∃𝑦(𝜓 ∧ (𝑦𝐶𝑥 = 𝐴)))
6 an12 533 . . . . . . . . . . 11 ((𝜓 ∧ (𝑦𝐶𝑥 = 𝐴)) ↔ (𝑦𝐶 ∧ (𝜓𝑥 = 𝐴)))
76exbii 1567 . . . . . . . . . 10 (∃𝑦(𝜓 ∧ (𝑦𝐶𝑥 = 𝐴)) ↔ ∃𝑦(𝑦𝐶 ∧ (𝜓𝑥 = 𝐴)))
85, 7sylib 121 . . . . . . . . 9 ((∀𝑦𝜓 ∧ ∃𝑦(𝑦𝐶𝑥 = 𝐴)) → ∃𝑦(𝑦𝐶 ∧ (𝜓𝑥 = 𝐴)))
9 df-rex 2397 . . . . . . . . 9 (∃𝑦𝐶 (𝜓𝑥 = 𝐴) ↔ ∃𝑦(𝑦𝐶 ∧ (𝜓𝑥 = 𝐴)))
108, 9sylibr 133 . . . . . . . 8 ((∀𝑦𝜓 ∧ ∃𝑦(𝑦𝐶𝑥 = 𝐴)) → ∃𝑦𝐶 (𝜓𝑥 = 𝐴))
114, 10sylan2b 283 . . . . . . 7 ((∀𝑦𝜓 ∧ ∃𝑦𝐶 𝑥 = 𝐴) → ∃𝑦𝐶 (𝜓𝑥 = 𝐴))
12 ralxfrd.3 . . . . . . . . . . 11 ((𝜑𝑥 = 𝐴) → (𝜓𝜒))
1312biimpd 143 . . . . . . . . . 10 ((𝜑𝑥 = 𝐴) → (𝜓𝜒))
1413expimpd 358 . . . . . . . . 9 (𝜑 → ((𝑥 = 𝐴𝜓) → 𝜒))
1514ancomsd 267 . . . . . . . 8 (𝜑 → ((𝜓𝑥 = 𝐴) → 𝜒))
1615reximdv 2508 . . . . . . 7 (𝜑 → (∃𝑦𝐶 (𝜓𝑥 = 𝐴) → ∃𝑦𝐶 𝜒))
1711, 16syl5 32 . . . . . 6 (𝜑 → ((∀𝑦𝜓 ∧ ∃𝑦𝐶 𝑥 = 𝐴) → ∃𝑦𝐶 𝜒))
1817adantr 272 . . . . 5 ((𝜑𝑥𝐵) → ((∀𝑦𝜓 ∧ ∃𝑦𝐶 𝑥 = 𝐴) → ∃𝑦𝐶 𝜒))
193, 18mpan2d 422 . . . 4 ((𝜑𝑥𝐵) → (∀𝑦𝜓 → ∃𝑦𝐶 𝜒))
202, 19syl5bir 152 . . 3 ((𝜑𝑥𝐵) → (𝜓 → ∃𝑦𝐶 𝜒))
2120rexlimdva 2524 . 2 (𝜑 → (∃𝑥𝐵 𝜓 → ∃𝑦𝐶 𝜒))
22 ralxfrd.1 . . . 4 ((𝜑𝑦𝐶) → 𝐴𝐵)
2312adantlr 466 . . . 4 (((𝜑𝑦𝐶) ∧ 𝑥 = 𝐴) → (𝜓𝜒))
2422, 23rspcedv 2765 . . 3 ((𝜑𝑦𝐶) → (𝜒 → ∃𝑥𝐵 𝜓))
2524rexlimdva 2524 . 2 (𝜑 → (∃𝑦𝐶 𝜒 → ∃𝑥𝐵 𝜓))
2621, 25impbid 128 1 (𝜑 → (∃𝑥𝐵 𝜓 ↔ ∃𝑦𝐶 𝜒))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104  wal 1312   = wceq 1314  wex 1451  wcel 1463  wrex 2392
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 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097
This theorem depends on definitions:  df-bi 116  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ral 2396  df-rex 2397  df-v 2660
This theorem is referenced by:  rexxfr2d  4354  rexxfr  4357
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