Theorem reximddv2 3020

Description: Double deduction from Theorem 19.22 of [Margaris] p. 90. (Contributed by Thierry Arnoux, 15-Dec-2019.)

Hypotheses

Ref	Expression
reximddv2.1	|- ( ( ( ( ph /\ x e. A ) /\ y e. B ) /\ ps ) -> ch )
reximddv2.2	|- ( ph -> E. x e. A E. y e. B ps )

Assertion

Ref	Expression
reximddv2	|- ( ph -> E. x e. A E. y e. B ch )

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

Allowed substitution hints:   ps( x, y)   ch( x, y)   A( x)   B( x, y)

Proof of Theorem reximddv2

Step	Hyp	Ref	Expression
1		reximddv2.1	. . . . 5 |- ( ( ( ( ph /\ x e. A ) /\ y e. B ) /\ ps ) -> ch )
2	1	ex 450	. . . 4 |- ( ( ( ph /\ x e. A ) /\ y e. B ) -> ( ps -> ch ) )
3	2	reximdva 3017	. . 3 |- ( ( ph /\ x e. A ) -> ( E. y e. B ps -> E. y e. B ch ) )
4	3	impr 649	. 2 |- ( ( ph /\ ( x e. A /\ E. y e. B ps ) ) -> E. y e. B ch )
5		reximddv2.2	. 2 |- ( ph -> E. x e. A E. y e. B ps )
6	4, 5	reximddv 3018	1 |- ( ph -> E. x e. A E. y e. B ch )

Syntax hints:   -> wi 4   /\ wa 384   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

This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1705  df-ral 2917  df-rex 2918

This theorem is referenced by:  prmgaplem8  15762  cpmadugsumfi  20682  cpmidg2sum  20685  cayhamlem4  20693  ltgseg  25491  cgraswap  25712  cgracom  25714  cgratr  25715  dfcgra2  25721  xrofsup  29533  prmunb2  38510