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Theorem rexrot4 2470
Description: Rotate existential restricted quantifiers twice. (Contributed by NM, 8-Apr-2015.)
Assertion
Ref Expression
rexrot4 (x A y B z 𝐶 w 𝐷 φz 𝐶 w 𝐷 x A y B φ)
Distinct variable groups:   z,w,A   w,B,z   x,w,y,𝐶   x,z,𝐷,y
Allowed substitution hints:   φ(x,y,z,w)   A(x,y)   B(x,y)   𝐶(z)   𝐷(w)

Proof of Theorem rexrot4
StepHypRef Expression
1 rexcom13 2469 . . 3 (y B z 𝐶 w 𝐷 φw 𝐷 z 𝐶 y B φ)
21rexbii 2325 . 2 (x A y B z 𝐶 w 𝐷 φx A w 𝐷 z 𝐶 y B φ)
3 rexcom13 2469 . 2 (x A w 𝐷 z 𝐶 y B φz 𝐶 w 𝐷 x A y B φ)
42, 3bitri 173 1 (x A y B z 𝐶 w 𝐷 φz 𝐶 w 𝐷 x A y B φ)
Colors of variables: wff set class
Syntax hints:  wb 98  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-cleq 2030  df-clel 2033  df-nfc 2164  df-rex 2306
This theorem is referenced by: (None)
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