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Theorem rexcom13 2469
Description: Swap 1st and 3rd restricted existential quantifiers. (Contributed by NM, 8-Apr-2015.)
Assertion
Ref Expression
rexcom13 (x A y B z 𝐶 φz 𝐶 y B x A φ)
Distinct variable groups:   y,z,A   x,z,B   x,y,𝐶
Allowed substitution hints:   φ(x,y,z)   A(x)   B(y)   𝐶(z)

Proof of Theorem rexcom13
StepHypRef Expression
1 rexcom 2468 . 2 (x A y B z 𝐶 φy B x A z 𝐶 φ)
2 rexcom 2468 . . 3 (x A z 𝐶 φz 𝐶 x A φ)
32rexbii 2325 . 2 (y B x A z 𝐶 φy B z 𝐶 x A φ)
4 rexcom 2468 . 2 (y B z 𝐶 x A φz 𝐶 y B x A φ)
51, 3, 43bitri 195 1 (x A y B z 𝐶 φz 𝐶 y B x A 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:  rexrot4  2470
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