Theorem 2reurmo 41182

Description: Double restricted quantification with restricted existential uniqueness and restricted "at most one.", analogous to 2eumo 2545. (Contributed by Alexander van der Vekens, 24-Jun-2017.)

Assertion

Ref	Expression
2reurmo	|- ( E! x e. A E* y e. B ph -> E* x e. A E! y e. B ph )

Distinct variable groups:   y, A   x, y   x, B

Allowed substitution hints:   ph( x, y)   A( x)   B( y)

Proof of Theorem 2reurmo

Step	Hyp	Ref	Expression
1		reuimrmo 41178	. 2 |- ( A. x e. A ( E! y e. B ph -> E* y e. B ph ) -> ( E! x e. A E* y e. B ph -> E* x e. A E! y e. B ph ) )
2		reurmo 3161	. . 3 |- ( E! y e. B ph -> E* y e. B ph )
3	2	a1i 11	. 2 |- ( x e. A -> ( E! y e. B ph -> E* y e. B ph ) )
4	1, 3	mprg 2926	1 |- ( E! x e. A E* y e. B ph -> E* x e. A E! y e. B ph )

Syntax hints:   -> wi 4   e. wcel 1990   E!wreu 2914   E*wrmo 2915

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-10 2019  ax-12 2047

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-ex 1705  df-nf 1710  df-eu 2474  df-mo 2475  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920

This theorem is referenced by: (None)