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Theorem rexxfrd 4161
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 ((φ y 𝐶) → A B)
ralxfrd.2 ((φ x B) → y 𝐶 x = A)
ralxfrd.3 ((φ x = A) → (ψχ))
Assertion
Ref Expression
rexxfrd (φ → (x B ψy 𝐶 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 1418 . . . . 5 yψ
2119.3 1443 . . . 4 (yψψ)
3 ralxfrd.2 . . . . 5 ((φ x B) → y 𝐶 x = A)
4 df-rex 2306 . . . . . . . 8 (y 𝐶 x = Ay(y 𝐶 x = A))
5 19.29 1508 . . . . . . . . . 10 ((yψ y(y 𝐶 x = A)) → y(ψ (y 𝐶 x = A)))
6 an12 495 . . . . . . . . . . 11 ((ψ (y 𝐶 x = A)) ↔ (y 𝐶 (ψ x = A)))
76exbii 1493 . . . . . . . . . 10 (y(ψ (y 𝐶 x = A)) ↔ y(y 𝐶 (ψ x = A)))
85, 7sylib 127 . . . . . . . . 9 ((yψ y(y 𝐶 x = A)) → y(y 𝐶 (ψ x = A)))
9 df-rex 2306 . . . . . . . . 9 (y 𝐶 (ψ x = A) ↔ y(y 𝐶 (ψ x = A)))
108, 9sylibr 137 . . . . . . . 8 ((yψ y(y 𝐶 x = A)) → y 𝐶 (ψ x = A))
114, 10sylan2b 271 . . . . . . 7 ((yψ y 𝐶 x = A) → y 𝐶 (ψ x = A))
12 ralxfrd.3 . . . . . . . . . . 11 ((φ x = A) → (ψχ))
1312biimpd 132 . . . . . . . . . 10 ((φ x = A) → (ψχ))
1413expimpd 345 . . . . . . . . 9 (φ → ((x = A ψ) → χ))
1514ancomsd 256 . . . . . . . 8 (φ → ((ψ x = A) → χ))
1615reximdv 2414 . . . . . . 7 (φ → (y 𝐶 (ψ x = A) → y 𝐶 χ))
1711, 16syl5 28 . . . . . 6 (φ → ((yψ y 𝐶 x = A) → y 𝐶 χ))
1817adantr 261 . . . . 5 ((φ x B) → ((yψ y 𝐶 x = A) → y 𝐶 χ))
193, 18mpan2d 404 . . . 4 ((φ x B) → (yψy 𝐶 χ))
202, 19syl5bir 142 . . 3 ((φ x B) → (ψy 𝐶 χ))
2120rexlimdva 2427 . 2 (φ → (x B ψy 𝐶 χ))
22 ralxfrd.1 . . . 4 ((φ y 𝐶) → A B)
2312adantlr 446 . . . 4 (((φ y 𝐶) x = A) → (ψχ))
2422, 23rspcedv 2654 . . 3 ((φ y 𝐶) → (χx B ψ))
2524rexlimdva 2427 . 2 (φ → (y 𝐶 χx B ψ))
2621, 25impbid 120 1 (φ → (x B ψy 𝐶 χ))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98  wal 1240   = wceq 1242  wex 1378   wcel 1390  wrex 2301
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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553
This theorem is referenced by:  rexxfr2d  4163  rexxfr  4166
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