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Theorem rexcom13 2773
 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 2772 . 2 (x A y B z C φy B x A z C φ)
2 rexcom 2772 . . 3 (x A z C φz C x A φ)
32rexbii 2639 . 2 (y B x A z C φy B z C x A φ)
4 rexcom 2772 . 2 (y B z C x A φz C y B x A φ)
51, 3, 43bitri 262 1 (x A y B z C φz C y B x A φ)
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 176  ∃wrex 2615 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-cleq 2346  df-clel 2349  df-nfc 2478  df-rex 2620 This theorem is referenced by:  rexrot4  2774
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