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Theorem ee4anv 1787
Description: Rearrange existential quantifiers. (Contributed by NM, 31-Jul-1995.)
Assertion
Ref Expression
ee4anv (xyzw(φ ψ) ↔ (xyφ zwψ))
Distinct variable groups:   φ,z   φ,w   ψ,x   ψ,y   y,z   x,w
Allowed substitution hints:   φ(x,y)   ψ(z,w)

Proof of Theorem ee4anv
StepHypRef Expression
1 excom 1532 . . 3 (yzw(φ ψ) ↔ zyw(φ ψ))
21exbii 1474 . 2 (xyzw(φ ψ) ↔ xzyw(φ ψ))
3 eeanv 1785 . . 3 (yw(φ ψ) ↔ (yφ wψ))
432exbii 1475 . 2 (xzyw(φ ψ) ↔ xz(yφ wψ))
5 eeanv 1785 . 2 (xz(yφ wψ) ↔ (xyφ zwψ))
62, 4, 53bitri 195 1 (xyzw(φ ψ) ↔ (xyφ zwψ))
Colors of variables: wff set class
Syntax hints:   wa 97  wb 98  wex 1358
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-5 1312  ax-7 1313  ax-gen 1314  ax-ie1 1359  ax-ie2 1360  ax-4 1377  ax-17 1396  ax-ial 1405
This theorem depends on definitions:  df-bi 110  df-nf 1326
This theorem is referenced by:  ee8anv  1788  cgsex4g  2564  th3qlem1  6115  dmaddpq  6232  dmmulpq  6233  ltdcnq  6250  enq0ref  6282  nqpnq0nq  6302  nqnq0a  6303  nqnq0m  6304  genpdisj  6372  axaddcl  6559  axmulcl  6561
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