Theorem exdistrv 1882

Description: Distribute a pair of existential quantifiers (over disjoint variables) over a conjunction. Combination of 19.41v 1874 and 19.42v 1878. For a version with fewer disjoint variable conditions but requiring more axioms, see eeanv 1904. (Contributed by BJ, 30-Sep-2022.)

Assertion

Ref	Expression
exdistrv	|- ( E. x E. y ( ph /\ ps ) <-> ( E. x ph /\ E. y ps ) )

Distinct variable groups:   ph, y   ps, x   x, y

Allowed substitution hints:   ph( x)   ps( y)

Proof of Theorem exdistrv

Step	Hyp	Ref	Expression
1		exdistr 1881	. 2 |- ( E. x E. y ( ph /\ ps ) <-> E. x ( ph /\ E. y ps ) )
2		19.41v 1874	. 2 |- ( E. x ( ph /\ E. y ps ) <-> ( E. x ph /\ E. y ps ) )
3	1, 2	bitri 183	1 |- ( E. x E. y ( ph /\ ps ) <-> ( E. x ph /\ E. y ps ) )

Syntax hints:   /\ wa 103   <-> wb 104   E.wex 1468

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1423  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-4 1487  ax-17 1506  ax-ial 1514

This theorem depends on definitions:  df-bi 116

This theorem is referenced by:  prodmodc  11350  txbasval  12439