Theorem rsp2e 3004

Description: Restricted specialization. (Contributed by FL, 4-Jun-2012.) (Proof shortened by Wolf Lammen, 7-Jan-2020.)

Assertion

Ref	Expression
rsp2e	|- ( ( x e. A /\ y e. B /\ ph ) -> E. x e. A E. y e. B ph )

Proof of Theorem rsp2e

Step	Hyp	Ref	Expression
1		rspe 3003	. . 3 |- ( ( y e. B /\ ph ) -> E. y e. B ph )
2		rspe 3003	. . 3 |- ( ( x e. A /\ E. y e. B ph ) -> E. x e. A E. y e. B ph )
3	1, 2	sylan2 491	. 2 |- ( ( x e. A /\ ( y e. B /\ ph ) ) -> E. x e. A E. y e. B ph )
4	3	3impb 1260	1 |- ( ( x e. A /\ y e. B /\ ph ) -> E. x e. A E. y e. B ph )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   e. wcel 1990   E.wrex 2913

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-12 2047

This theorem depends on definitions:  df-bi 197  df-an 386  df-3an 1039  df-ex 1705  df-rex 2918

This theorem is referenced by:  pell14qrdich  37433